{"id":3179,"date":"2022-01-27T11:48:57","date_gmt":"2022-01-27T06:18:57","guid":{"rendered":"https:\/\/soulofmathematics.com\/?page_id=3179"},"modified":"2022-01-27T11:50:41","modified_gmt":"2022-01-27T06:20:41","slug":"topological-spaces","status":"publish","type":"page","link":"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/","title":{"rendered":"TOPOLOGICAL SPACES"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"3179\" class=\"elementor elementor-3179\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4e75cca elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4e75cca\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2a9ca50\" data-id=\"2a9ca50\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b0ade07 elementor-widget elementor-widget-image\" data-id=\"b0ade07\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"582\" height=\"253\" src=\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png?fit=582%2C253&amp;ssl=1\" class=\"elementor-animation-grow attachment-large size-large wp-image-3204\" alt=\"\" srcset=\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png?w=582&amp;ssl=1 582w, https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png?resize=300%2C130&amp;ssl=1 300w, https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png?resize=150%2C65&amp;ssl=1 150w\" sizes=\"(max-width: 582px) 100vw, 582px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4c81367 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4c81367\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1d35088\" data-id=\"1d35088\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-28f9194 elementor-invisible elementor-widget elementor-widget-eael-fancy-text\" data-id=\"28f9194\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeIn&quot;}\" data-widget_type=\"eael-fancy-text.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t<div  class=\"eael-fancy-text-container style-1\" data-fancy-text-id=\"28f9194\" data-fancy-text=\"|A Topological Space|is,|A Geometrical Space in which Closeness is defined|but,|cannot necessarily be measured by a Numeric Distance.\" data-fancy-text-transition-type=\"typing\" data-fancy-text-speed=\"50\" data-fancy-text-delay=\"2500\" data-fancy-text-cursor=\"yes\" data-fancy-text-loop=\"yes\" data-fancy-text-action=\"page_load\" >\n\t\t\n\t\t\n\t\t\t\t\t<span id=\"eael-fancy-text-28f9194\" class=\"eael-fancy-text-strings \">\n\t\t\t\t<noscript>\n\t\t\t\t\tA Topological Space, is,, A Geometrical Space in which Closeness is defined, but,, cannot necessarily be measured by a Numeric Distance.\t\t\t\t<\/noscript>\n\t\t\t<\/span>\n\t\t\n\t\t\t<\/div><!-- close .eael-fancy-text-container -->\n\n\t<div class=\"clearfix\"><\/div>\n\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-52271bd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"52271bd\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8ef7021\" data-id=\"8ef7021\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-998a66b elementor-widget elementor-widget-spacer\" data-id=\"998a66b\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1ccdfab elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1ccdfab\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9e91300\" data-id=\"9e91300\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-b418c9c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b418c9c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-1a3226c\" data-id=\"1a3226c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-30a2303 elementor-widget elementor-widget-text-editor\" data-id=\"30a2303\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>More specifically, a topological space is a\u00a0set\u00a0of\u00a0points, along with a set of\u00a0neighborhoods\u00a0for each point, satisfying a set of\u00a0<a title=\"Axiom\" href=\"https:\/\/en.wikipedia.org\/wiki\/Axiom#Non-logical_axioms\">axioms<\/a> relating points and neighborhoods. A topological space is the most general type of a\u00a0<a title=\"Space (mathematics)\" href=\"https:\/\/en.wikipedia.org\/wiki\/Space_(mathematics)\">mathematical space<\/a>\u00a0that allows for the definition of\u00a0<a title=\"Limit (mathematics)\" href=\"https:\/\/en.wikipedia.org\/wiki\/Limit_(mathematics)\">limits<\/a>,\u00a0<a class=\"mw-redirect\" title=\"Continuous function (topology)\" href=\"https:\/\/en.wikipedia.org\/wiki\/Continuous_function_(topology)\">continuity<\/a>, and\u00a0connectedness.<sup id=\"cite_ref-1\" class=\"reference\"><\/sup>\u00a0Other spaces, such as\u00a0<a title=\"Euclidean space\" href=\"https:\/\/en.wikipedia.org\/wiki\/Euclidean_space\">Euclidean spaces<\/a>,\u00a0<a title=\"Metric space\" href=\"https:\/\/en.wikipedia.org\/wiki\/Metric_space\">metric spaces<\/a>\u00a0and\u00a0<a title=\"Manifold\" href=\"https:\/\/en.wikipedia.org\/wiki\/Manifold\">manifolds<\/a>, are topological spaces with extra\u00a0structures, properties or constraints.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-532b39f\" data-id=\"532b39f\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-eec3712 elementor-widget elementor-widget-text-editor\" data-id=\"eec3712\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><strong>Definition<\/strong>\u2013 A set\u00a0<em>X<\/em>\u00a0for which a <a href=\"https:\/\/soulofmathematics.com\/index.php\/the-topology-series\/\">topology\u00a0<em>T<\/em><\/a>\u00a0has been specified is called a topological space.<\/p><p>A topological space is an ordered pair (<em>X, T<\/em>) consisting of a set\u00a0<em>X<\/em>\u00a0and a topology\u00a0<em>T<\/em>\u00a0on\u00a0<em>X<\/em>.<\/p><p>If\u00a0<em>X<\/em>\u00a0is a topological space with topology\u00a0<em>T<\/em>, we say that a subset\u00a0<em>U<\/em>\u00a0of\u00a0<em>X<\/em>\u00a0is an open set of\u00a0<em>X<\/em>\u00a0if\u00a0<em>U<\/em>\u00a0belongs to collection of\u00a0<em>T<\/em>.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1e4f8ba elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1e4f8ba\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-aaabebe\" data-id=\"aaabebe\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-de8f877 elementor-widget elementor-widget-image\" data-id=\"de8f877\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"720\" height=\"638\" src=\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/720px-Topological_space_examples.svg.png?fit=720%2C638&amp;ssl=1\" class=\"elementor-animation-grow attachment-large size-large wp-image-3205\" alt=\"\" srcset=\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/720px-Topological_space_examples.svg.png?w=720&amp;ssl=1 720w, https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/720px-Topological_space_examples.svg.png?resize=300%2C266&amp;ssl=1 300w, https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/720px-Topological_space_examples.svg.png?resize=150%2C133&amp;ssl=1 150w\" sizes=\"(max-width: 720px) 100vw, 720px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5748f3e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5748f3e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-484f03e\" data-id=\"484f03e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-511872e elementor-widget elementor-widget-text-editor\" data-id=\"511872e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Example- Let (X, T) be a topological space and Y a subset of X. Then,<\/p><p>S = { H \u2282 Y | H = G <em>\u2229 Y <\/em>for some<em> G <strong>\u2208 <\/strong>T<\/em>}<\/p><p>is a topology on Y. The open sets in Y are the intersections of the open sets in X with Y. This topology is called the induced or relative topology of Y in X, and (Y, S) is called a topological subspace of (X, T). For instance, the interval [0,1\/2) is an open open subset of [0, 1] with respect to the induced metric topology of [0, 1] in <strong>R<\/strong>, since [0,1\/2) = (-1\/2, 1\/2) <em>\u2229\u00a0<\/em>[0, 1].<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c639f89 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c639f89\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5de40bd\" data-id=\"5de40bd\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0408480 elementor-widget elementor-widget-text-editor\" data-id=\"0408480\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>A set V \u2282 X is a neighborhood of a point x <em><strong>\u2208<\/strong><\/em> X if there exists an open set G \u2282 V with x <em><strong>\u2208 <\/strong><\/em>G. We do not require that V itself is open. A topology T on X is called Hausdorff if every pair of distinct points x, y <em><strong>\u2208\u00a0<\/strong><\/em>X has a pair of non-intersecting neighborhoods, meaning that there are neighborhoods V<sub>x\u00a0<\/sub><span style=\"letter-spacing: 0px;\">of x and V<sub>y <\/sub>of y such that V<sub>x <\/sub><em>\u2229<\/em> V<sub>x\u00a0<\/sub>= \u03d5. When the topology is clear, we often refer to X as a Hausdorff space. Almost all the topological spaces encountered in analysis are Hausdorff. For example, all metric topologies are Hausdorff.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0321e27 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0321e27\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-bc23b80\" data-id=\"bc23b80\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5190fc3 elementor-widget elementor-widget-text-editor\" data-id=\"5190fc3\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Definition- A sequence (x<sub>n<\/sub>) in X converges to a limit x\u00a0<em><strong>\u2208<\/strong><\/em> X if for every neighborhood V of x, there is a number N such that x<sub>n<\/sub><em><strong>\u2208<\/strong><\/em> V for all n \u2265 N.<\/p><p>This definition says that the sequence eventually lies entirely in every neighborhood of x.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-447231a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"447231a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ae3f430\" data-id=\"ae3f430\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c028dc5 elementor-widget elementor-widget-text-editor\" data-id=\"c028dc5\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Definition- A function f : X \u2192 Y is continuous at x <em><strong>\u2208 <\/strong><\/em>X if for each neighborhood W of f(x) there exists a neighborhood V of x such that f(V) \u2282 W. We say that f is continuous on X if it is continuous at every x <em><strong>\u2208 <\/strong><\/em>X.<\/p><p>\u00a0<\/p><p>Theorem- Let (X, T) and (Y, S) be two topological spaces and f : X \u2192 Y.<\/p><p>Then f is continuous on X if and only if f<sup>-1 <\/sup>(G) <em><strong>\u2208 <\/strong><\/em>T for every G <em><strong>\u2208 <\/strong><\/em>S.<\/p><p>\u00a0<\/p><p>Definition- A function f : X \u2192 Y between topological spaces X and Y is a homeomorphism if it is a one to one, onto map and both f and f<sup>-1 <\/sup>are continuous. Two topological spaces X and Y are homeomorphic if there is a homeomorphism f : X \u2192 Y.<\/p><p>\u00a0<\/p><p>Homeomorphic spaces are indistinguishable as topological spaces. For example, if f : X \u2192 Y is a homeomorphism, then G is open in X if f(G) is open in Y, and a sequence (x<sub>n<\/sub>) converges to x in X if and only if the sequence (f(x<sub>n<\/sub>)) converges to f(x) in Y.<\/p><p>A one to one, onto map f always has an inverse f<sup>-1<\/sup>, but f<sup>-1\u00a0<\/sup>need not be continuous even if f is.<\/p><p>\u00a0<\/p><p>\u00a0<\/p><p>Example- We define f : [0, 2\u03c0) \u2192 T by f(\u0398) = e<sup>i\u0398<\/sup>, where [0, 2\u03c0) \u2282 R with the topology induced by the usual topology on R, and T \u2282 C is the unit circle with the topology induced by the usual topology on C. Then, ass illustrated in figure below, f is continuous but f<sup>-1 <\/sup>is not.\u00a0<\/p><p>\u00a0<\/p><p>\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-70636b2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"70636b2\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5f43dc1\" data-id=\"5f43dc1\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-487b5b7 elementor-widget elementor-widget-image\" data-id=\"487b5b7\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"960\" height=\"393\" src=\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/sc.png?fit=960%2C393&amp;ssl=1\" class=\"elementor-animation-grow attachment-large size-large wp-image-3225\" alt=\"The interval and the circle are not homeomorphic.\" srcset=\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/sc.png?w=1254&amp;ssl=1 1254w, https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/sc.png?resize=300%2C123&amp;ssl=1 300w, https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/sc.png?resize=1024%2C419&amp;ssl=1 1024w, https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/sc.png?resize=768%2C314&amp;ssl=1 768w, https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/sc.png?resize=1140%2C466&amp;ssl=1 1140w, https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/sc.png?resize=150%2C61&amp;ssl=1 150w\" sizes=\"(max-width: 960px) 100vw, 960px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>A Topological Space, is,, A Geometrical Space in which Closeness is defined, but,, cannot necessarily be measured by a Numeric Distance. More specifically, a topological space is a\u00a0set\u00a0of\u00a0points, along with a set of\u00a0neighborhoods\u00a0for each point, satisfying a set of\u00a0axioms relating points and neighborhoods. A topological space is the most general type of a\u00a0mathematical space\u00a0that allows for the definition of\u00a0limits,\u00a0continuity, and\u00a0connectedness.\u00a0Other spaces, such as\u00a0Euclidean spaces,\u00a0metric spaces\u00a0and\u00a0manifolds, are topological spaces with extra\u00a0structures, properties or constraints. Definition\u2013 A set\u00a0X\u00a0for which a topology\u00a0T\u00a0has been specified is called a topological space. A topological space is an ordered pair (X, T) consisting of a set\u00a0X\u00a0and a topology\u00a0T\u00a0on\u00a0X. If\u00a0X\u00a0is a topological space with topology\u00a0T, we say that a subset\u00a0U\u00a0of\u00a0X\u00a0is an open set of\u00a0X\u00a0if\u00a0U\u00a0belongs to collection of\u00a0T. Example- Let (X, T) be a topological space and Y a subset of X. Then, S = { H \u2282 Y | H = G \u2229 Y for some G \u2208 T} is a topology on Y. The open sets in Y are the intersections of the open sets in X with Y. This topology is called the induced or relative topology of Y in X, and (Y, S) is called a topological subspace of (X, T). For instance, the interval [0,1\/2) is an open open subset of [0, 1] with respect to the induced metric topology of [0, 1] in R, since [0,1\/2) = (-1\/2, 1\/2) \u2229\u00a0[0, 1]. A set V \u2282 X is a neighborhood of a point x \u2208 X if there exists an open set G \u2282 V with x \u2208 G. We do not require that V itself is open. A topology T on X is called Hausdorff if every pair of distinct points x, y \u2208\u00a0X has a pair of non-intersecting neighborhoods, meaning that there are neighborhoods Vx\u00a0of x and Vy of y such that Vx \u2229 Vx\u00a0= \u03d5. When the topology is clear, we often refer to X as a Hausdorff space. Almost all the topological spaces encountered in analysis are Hausdorff. For example, all metric topologies are Hausdorff. Definition- A sequence (xn) in X converges to a limit x\u00a0\u2208 X if for every neighborhood V of x, there is a number N such that xn\u2208 V for all n \u2265 N. This definition says that the sequence eventually lies entirely in every neighborhood of x. Definition- A function f : X \u2192 Y is continuous at x \u2208 X if for each neighborhood W of f(x) there exists a neighborhood V of x such that f(V) \u2282 W. We say that f is continuous on X if it is continuous at every x \u2208 X. \u00a0 Theorem- Let (X, T) and (Y, S) be two topological spaces and f : X \u2192 Y. Then f is continuous on X if and only if f-1 (G) \u2208 T for every G \u2208 S. \u00a0 Definition- A function f : X \u2192 Y between topological spaces X and Y is a homeomorphism if it is a one to one, onto map and both f and f-1 are continuous. Two topological spaces X and Y are homeomorphic if there is a homeomorphism f : X \u2192 Y. \u00a0 Homeomorphic spaces are indistinguishable as topological spaces. For example, if f : X \u2192 Y is a homeomorphism, then G is open in X if f(G) is open in Y, and a sequence (xn) converges to x in X if and only if the sequence (f(xn)) converges to f(x) in Y. A one to one, onto map f always has an inverse f-1, but f-1\u00a0need not be continuous even if f is. \u00a0 \u00a0 Example- We define f : [0, 2\u03c0) \u2192 T by f(\u0398) = ei\u0398, where [0, 2\u03c0) \u2282 R with the topology induced by the usual topology on R, and T \u2282 C is the unit circle with the topology induced by the usual topology on C. Then, ass illustrated in figure below, f is continuous but f-1 is not.\u00a0 \u00a0 \u00a0<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false,"footnotes":""},"class_list":["post-3179","page","type-page","status-publish","hentry"],"ams_acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>TOPOLOGICAL SPACES - SOUL OF MATHEMATICS<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"TOPOLOGICAL SPACES - SOUL OF MATHEMATICS\" \/>\n<meta property=\"og:description\" content=\"A Topological Space, is,, A Geometrical Space in which Closeness is defined, but,, cannot necessarily be measured by a Numeric Distance. More specifically, a topological space is a\u00a0set\u00a0of\u00a0points, along with a set of\u00a0neighborhoods\u00a0for each point, satisfying a set of\u00a0axioms relating points and neighborhoods. A topological space is the most general type of a\u00a0mathematical space\u00a0that allows for the definition of\u00a0limits,\u00a0continuity, and\u00a0connectedness.\u00a0Other spaces, such as\u00a0Euclidean spaces,\u00a0metric spaces\u00a0and\u00a0manifolds, are topological spaces with extra\u00a0structures, properties or constraints. Definition\u2013 A set\u00a0X\u00a0for which a topology\u00a0T\u00a0has been specified is called a topological space. A topological space is an ordered pair (X, T) consisting of a set\u00a0X\u00a0and a topology\u00a0T\u00a0on\u00a0X. If\u00a0X\u00a0is a topological space with topology\u00a0T, we say that a subset\u00a0U\u00a0of\u00a0X\u00a0is an open set of\u00a0X\u00a0if\u00a0U\u00a0belongs to collection of\u00a0T. Example- Let (X, T) be a topological space and Y a subset of X. Then, S = { H \u2282 Y | H = G \u2229 Y for some G \u2208 T} is a topology on Y. The open sets in Y are the intersections of the open sets in X with Y. This topology is called the induced or relative topology of Y in X, and (Y, S) is called a topological subspace of (X, T). For instance, the interval [0,1\/2) is an open open subset of [0, 1] with respect to the induced metric topology of [0, 1] in R, since [0,1\/2) = (-1\/2, 1\/2) \u2229\u00a0[0, 1]. A set V \u2282 X is a neighborhood of a point x \u2208 X if there exists an open set G \u2282 V with x \u2208 G. We do not require that V itself is open. A topology T on X is called Hausdorff if every pair of distinct points x, y \u2208\u00a0X has a pair of non-intersecting neighborhoods, meaning that there are neighborhoods Vx\u00a0of x and Vy of y such that Vx \u2229 Vx\u00a0= \u03d5. When the topology is clear, we often refer to X as a Hausdorff space. Almost all the topological spaces encountered in analysis are Hausdorff. For example, all metric topologies are Hausdorff. Definition- A sequence (xn) in X converges to a limit x\u00a0\u2208 X if for every neighborhood V of x, there is a number N such that xn\u2208 V for all n \u2265 N. This definition says that the sequence eventually lies entirely in every neighborhood of x. Definition- A function f : X \u2192 Y is continuous at x \u2208 X if for each neighborhood W of f(x) there exists a neighborhood V of x such that f(V) \u2282 W. We say that f is continuous on X if it is continuous at every x \u2208 X. \u00a0 Theorem- Let (X, T) and (Y, S) be two topological spaces and f : X \u2192 Y. Then f is continuous on X if and only if f-1 (G) \u2208 T for every G \u2208 S. \u00a0 Definition- A function f : X \u2192 Y between topological spaces X and Y is a homeomorphism if it is a one to one, onto map and both f and f-1 are continuous. Two topological spaces X and Y are homeomorphic if there is a homeomorphism f : X \u2192 Y. \u00a0 Homeomorphic spaces are indistinguishable as topological spaces. For example, if f : X \u2192 Y is a homeomorphism, then G is open in X if f(G) is open in Y, and a sequence (xn) converges to x in X if and only if the sequence (f(xn)) converges to f(x) in Y. A one to one, onto map f always has an inverse f-1, but f-1\u00a0need not be continuous even if f is. \u00a0 \u00a0 Example- We define f : [0, 2\u03c0) \u2192 T by f(\u0398) = ei\u0398, where [0, 2\u03c0) \u2282 R with the topology induced by the usual topology on R, and T \u2282 C is the unit circle with the topology induced by the usual topology on C. Then, ass illustrated in figure below, f is continuous but f-1 is not.\u00a0 \u00a0 \u00a0\" \/>\n<meta property=\"og:url\" content=\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/\" \/>\n<meta property=\"og:site_name\" content=\"SOUL OF MATHEMATICS\" \/>\n<meta property=\"article:modified_time\" content=\"2022-01-27T06:20:41+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/\",\"url\":\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/\",\"name\":\"TOPOLOGICAL SPACES - SOUL OF MATHEMATICS\",\"isPartOf\":{\"@id\":\"https:\/\/soulofmathematics.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png\",\"datePublished\":\"2022-01-27T06:18:57+00:00\",\"dateModified\":\"2022-01-27T06:20:41+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/#primaryimage\",\"url\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png?fit=582%2C253&ssl=1\",\"contentUrl\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png?fit=582%2C253&ssl=1\",\"width\":582,\"height\":253},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/soulofmathematics.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"TOPOLOGICAL SPACES\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/soulofmathematics.com\/#website\",\"url\":\"https:\/\/soulofmathematics.com\/\",\"name\":\"SOUL OF MATHEMATICS\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/soulofmathematics.com\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"TOPOLOGICAL SPACES - SOUL OF MATHEMATICS","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/","og_locale":"en_US","og_type":"article","og_title":"TOPOLOGICAL SPACES - SOUL OF MATHEMATICS","og_description":"A Topological Space, is,, A Geometrical Space in which Closeness is defined, but,, cannot necessarily be measured by a Numeric Distance. More specifically, a topological space is a\u00a0set\u00a0of\u00a0points, along with a set of\u00a0neighborhoods\u00a0for each point, satisfying a set of\u00a0axioms relating points and neighborhoods. A topological space is the most general type of a\u00a0mathematical space\u00a0that allows for the definition of\u00a0limits,\u00a0continuity, and\u00a0connectedness.\u00a0Other spaces, such as\u00a0Euclidean spaces,\u00a0metric spaces\u00a0and\u00a0manifolds, are topological spaces with extra\u00a0structures, properties or constraints. Definition\u2013 A set\u00a0X\u00a0for which a topology\u00a0T\u00a0has been specified is called a topological space. A topological space is an ordered pair (X, T) consisting of a set\u00a0X\u00a0and a topology\u00a0T\u00a0on\u00a0X. If\u00a0X\u00a0is a topological space with topology\u00a0T, we say that a subset\u00a0U\u00a0of\u00a0X\u00a0is an open set of\u00a0X\u00a0if\u00a0U\u00a0belongs to collection of\u00a0T. Example- Let (X, T) be a topological space and Y a subset of X. Then, S = { H \u2282 Y | H = G \u2229 Y for some G \u2208 T} is a topology on Y. The open sets in Y are the intersections of the open sets in X with Y. This topology is called the induced or relative topology of Y in X, and (Y, S) is called a topological subspace of (X, T). For instance, the interval [0,1\/2) is an open open subset of [0, 1] with respect to the induced metric topology of [0, 1] in R, since [0,1\/2) = (-1\/2, 1\/2) \u2229\u00a0[0, 1]. A set V \u2282 X is a neighborhood of a point x \u2208 X if there exists an open set G \u2282 V with x \u2208 G. We do not require that V itself is open. A topology T on X is called Hausdorff if every pair of distinct points x, y \u2208\u00a0X has a pair of non-intersecting neighborhoods, meaning that there are neighborhoods Vx\u00a0of x and Vy of y such that Vx \u2229 Vx\u00a0= \u03d5. When the topology is clear, we often refer to X as a Hausdorff space. Almost all the topological spaces encountered in analysis are Hausdorff. For example, all metric topologies are Hausdorff. Definition- A sequence (xn) in X converges to a limit x\u00a0\u2208 X if for every neighborhood V of x, there is a number N such that xn\u2208 V for all n \u2265 N. This definition says that the sequence eventually lies entirely in every neighborhood of x. Definition- A function f : X \u2192 Y is continuous at x \u2208 X if for each neighborhood W of f(x) there exists a neighborhood V of x such that f(V) \u2282 W. We say that f is continuous on X if it is continuous at every x \u2208 X. \u00a0 Theorem- Let (X, T) and (Y, S) be two topological spaces and f : X \u2192 Y. Then f is continuous on X if and only if f-1 (G) \u2208 T for every G \u2208 S. \u00a0 Definition- A function f : X \u2192 Y between topological spaces X and Y is a homeomorphism if it is a one to one, onto map and both f and f-1 are continuous. Two topological spaces X and Y are homeomorphic if there is a homeomorphism f : X \u2192 Y. \u00a0 Homeomorphic spaces are indistinguishable as topological spaces. For example, if f : X \u2192 Y is a homeomorphism, then G is open in X if f(G) is open in Y, and a sequence (xn) converges to x in X if and only if the sequence (f(xn)) converges to f(x) in Y. A one to one, onto map f always has an inverse f-1, but f-1\u00a0need not be continuous even if f is. \u00a0 \u00a0 Example- We define f : [0, 2\u03c0) \u2192 T by f(\u0398) = ei\u0398, where [0, 2\u03c0) \u2282 R with the topology induced by the usual topology on R, and T \u2282 C is the unit circle with the topology induced by the usual topology on C. Then, ass illustrated in figure below, f is continuous but f-1 is not.\u00a0 \u00a0 \u00a0","og_url":"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/","og_site_name":"SOUL OF MATHEMATICS","article_modified_time":"2022-01-27T06:20:41+00:00","og_image":[{"url":"https:\/\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png"}],"twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/","url":"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/","name":"TOPOLOGICAL SPACES - SOUL OF MATHEMATICS","isPartOf":{"@id":"https:\/\/soulofmathematics.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/#primaryimage"},"image":{"@id":"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/#primaryimage"},"thumbnailUrl":"https:\/\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png","datePublished":"2022-01-27T06:18:57+00:00","dateModified":"2022-01-27T06:20:41+00:00","breadcrumb":{"@id":"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/#primaryimage","url":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png?fit=582%2C253&ssl=1","contentUrl":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png?fit=582%2C253&ssl=1","width":582,"height":253},{"@type":"BreadcrumbList","@id":"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/soulofmathematics.com\/"},{"@type":"ListItem","position":2,"name":"TOPOLOGICAL SPACES"}]},{"@type":"WebSite","@id":"https:\/\/soulofmathematics.com\/#website","url":"https:\/\/soulofmathematics.com\/","name":"SOUL OF MATHEMATICS","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/soulofmathematics.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"}]}},"jetpack_sharing_enabled":true,"jetpack-related-posts":[],"jetpack_shortlink":"https:\/\/wp.me\/Pcfs4y-Ph","_links":{"self":[{"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/pages\/3179","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/comments?post=3179"}],"version-history":[{"count":42,"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/pages\/3179\/revisions"}],"predecessor-version":[{"id":3240,"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/pages\/3179\/revisions\/3240"}],"wp:attachment":[{"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/media?parent=3179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}