{"version":"1.0","provider_name":"SOUL OF MATHEMATICS","provider_url":"https:\/\/soulofmathematics.com","author_name":"Rajarshi Dey","author_url":"https:\/\/soulofmathematics.com\/index.php\/author\/rajarshidey1729gmail-com\/","title":"TOPOLOGICAL SPACES - SOUL OF MATHEMATICS","type":"rich","width":600,"height":338,"html":"<blockquote class=\"wp-embedded-content\" data-secret=\"efAsxuZrWx\"><a href=\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/\">TOPOLOGICAL SPACES<\/a><\/blockquote><iframe sandbox=\"allow-scripts\" security=\"restricted\" src=\"https:\/\/soulofmathematics.com\/index.php\/topological-spaces\/embed\/#?secret=efAsxuZrWx\" width=\"600\" height=\"338\" title=\"&#8220;TOPOLOGICAL SPACES&#8221; &#8212; SOUL OF MATHEMATICS\" data-secret=\"efAsxuZrWx\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\" class=\"wp-embedded-content\"><\/iframe><script type=\"text\/javascript\">\n\/* <![CDATA[ *\/\n\/*! This file is auto-generated *\/\n!function(d,l){\"use strict\";l.querySelector&&d.addEventListener&&\"undefined\"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!\/[^a-zA-Z0-9]\/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret=\"'+t.secret+'\"]'),o=l.querySelectorAll('blockquote[data-secret=\"'+t.secret+'\"]'),c=new RegExp(\"^https?:$\",\"i\"),i=0;i<o.length;i++)o[i].style.display=\"none\";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute(\"style\"),\"height\"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):\"link\"===t.message&&(r=new URL(s.getAttribute(\"src\")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener(\"message\",d.wp.receiveEmbedMessage,!1),l.addEventListener(\"DOMContentLoaded\",function(){for(var e,t,s=l.querySelectorAll(\"iframe.wp-embedded-content\"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute(\"data-secret\"))||(t=Math.random().toString(36).substring(2,12),e.src+=\"#?secret=\"+t,e.setAttribute(\"data-secret\",t)),e.contentWindow.postMessage({message:\"ready\",secret:t},\"*\")},!1)))}(window,document);\n\/* ]]> *\/\n<\/script>\n","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","thumbnail_url":"https:\/\/soulofmathematics.com\/wp-content\/uploads\/2022\/01\/compact_spaces-removebg-preview-1.png"}