{"id":3075,"date":"2021-12-05T13:12:01","date_gmt":"2021-12-05T07:42:01","guid":{"rendered":"https:\/\/soulofmathematics.com\/?page_id=3075"},"modified":"2021-12-06T10:33:47","modified_gmt":"2021-12-06T05:03:47","slug":"the-topology-series","status":"publish","type":"page","link":"https:\/\/soulofmathematics.com\/index.php\/the-topology-series\/","title":{"rendered":"The Topology Series"},"content":{"rendered":"\n
\"\"<\/figure>
\n

Chapter 1<\/h2>\n

Prerequisites- Set Theory<\/p>\n

<\/p>\n\n\n\n<\/p>\n

What is topology?<\/h3>\n

\n\n\n\n<\/p>\n

In simple terms, Topology is concerned with the geometric properties of objects when it undergoes physical distortions like stretching, twisting, and bending.<\/p>\n

\n\n\n\n<\/p>\n

Now we dive a bit deeper in terms of mathematics.<\/p>\n

\n\n\n\n<\/p>\n

\n\n\n\n<\/p>\n

\u00a0<\/p>\n<\/div><\/div>\n\n\n\n

<\/span>
\n

<\/p>\n\n\n\n<\/p>\n

Definition 1<\/strong>– A topology on a set X<\/em> is a collection T<\/em> of subsets of X<\/em> having the following properties-<\/p>\n

    \n
  1. \u03a6<\/em> and X<\/em> are in T<\/em>.<\/li>\n
  2. The union of the elements of any subcollection of T<\/em> is in T<\/em>.<\/li>\n
  3. The intersection of the elements of any finite subcollection of T<\/em> is in T<\/em>.<\/li>\n<\/ol>\n\n\n\n

    Definition 2<\/strong>– A set X<\/em> for which a topology T<\/em> has been specified is called a topological space.<\/p>\n\n\n\n

    A topological space is an ordered pair (X, T<\/em>) consisting of a set X<\/em> and a topology T<\/em> on X<\/em>.<\/p>\n\n\n\n

    If X<\/em> is a topological space with topology T<\/em>, we say that a subset U<\/em> of X<\/em> is an open set of X<\/em> if U<\/em> belongs to collection of T<\/em>.<\/p>\n\n\n\n

    A topological space is a set X<\/em> are both open, and such that arbitrary unions and finite intersections of open sets are open.<\/p>\n\n\n\n

    If X<\/em> is any set, the collection of all subsets of X<\/em>, it is called the discrete Topology. The collection consisting of X<\/em>, we shall call it the indiscrete Topology, or the trivial Topology.<\/p>\n\n\n\n

    Example<\/strong>– Let X<\/em> be a set, let Tf<\/sub><\/em> be the collection of all subsets U<\/em> of X<\/em> such that X-U<\/em> either is finite or is all of X<\/em>. Then Tf<\/sub> is a topology on X<\/em>, called the finite complement topology. Both X<\/em> and \u03a6<\/em> are in Tf<\/sub><\/em>, since X-X<\/em> is finite and X-\u03a6<\/em> is all of X<\/em>. If {U\u03b1<\/sub><\/em>} is an indexed family of non- empty elements of Tf<\/sub><\/em>, to show that \u222aU\u03b1<\/sub><\/em>, is in Tf<\/sub><\/em>, we compute,<\/p>\n\n\n\n

    X – \u222aU\u03b1<\/sub> = \u2229(X – U\u03b1<\/sub>)<\/em>.<\/p>\n\n\n\n

    The set is finite because each set X – U\u03b1<\/sub><\/em> is finite. U1<\/sub>, U2<\/sub>,……….,Un<\/sub><\/em> are non-empty elements of Tf<\/sub><\/em>, to show that \u2229U1<\/sub><\/em> is in Tf<\/sub><\/em>, we compute,<\/p>\n\n\n\n

    X – \u2229<\/em>Ui<\/sub> = \u222a<\/em>(X – Ui<\/sub>)<\/em> (i = 1, to n).<\/p>\n<\/div><\/div>\n\n\n\n

    Point set topology<\/strong><\/strong> is a disease from which the human race will soon recover.<\/p>Henri Poincare<\/em><\/cite><\/blockquote><\/figure>\n\n\n\n

    <\/span>
    \n

    Definition 3<\/strong>– Suppose that T<\/em> and T’<\/em> are two topologies on a given set X<\/em>. If T<\/em>\u2282T’<\/em>, we say that T’<\/em> is finer than T<\/em>. We also say T<\/em> is coarser than T’<\/em>, or strictly coarser, in these two respective situations. Wes say that T<\/em> is comparable with T’<\/em> if either T<\/em>\u2282T’<\/em> or T<\/em>‘\u2282T<\/em>.<\/p>\n\n\n\n

    Definition 4<\/strong>– If X<\/em> is a set, a basis for a topology on X<\/em> is a collection B<\/em><\/strong> of subsets X<\/em> (called basis elements) such that-<\/p>\n\n\n\n

    1. For each x<\/em> \u2208<\/strong> X<\/em>, there is at least one basis element B<\/em> containing x<\/em>.<\/li>
    2. If x belongs to the intersection of two basis elements B<\/em>1<\/em><\/sub> and B<\/em>2<\/em><\/sub>, then there is a basis element B<\/em>3<\/em><\/sub> containing x such that B<\/em>3<\/em><\/sub> \u2282 B<\/em>1<\/em><\/sub>\u2229<\/em><\/em>B<\/em>2<\/em><\/sub>.<\/li><\/ol>\n\n\n\n

      If B<\/em><\/strong> satisfies these two conditions, then we define the topology T<\/em> generated by B<\/em><\/strong> as follows-<\/p>\n\n\n\n

      A subset U<\/em> of X<\/em> is said to be open in X<\/em> (that is, to be an element of T<\/em>) if for each x<\/em> \u2208<\/strong> U<\/em>, there is a basis element B \u2208<\/strong> B<\/em><\/strong> such that x<\/em> \u2208<\/strong> B<\/em><\/strong> and B<\/em><\/strong> \u2282 U<\/em>.<\/p>\n\n\n\n

      Note that each basis element is itself an element of T<\/em>.<\/p>\n\n\n\n

      To understand better we will go through a few lemmas.<\/p><\/blockquote>\n\n\n\n

      Lemma 1<\/strong>– Let X<\/em> be a set; let B<\/em><\/strong> be a basis for topology T<\/em> on X<\/em>. Then T<\/em> equals the collection of all unions of elements of B<\/em><\/strong>.<\/p>\n\n\n\n

      Proof- Given a collection of elements of B<\/em><\/strong>, they are also elements of T<\/em>. Because T<\/em> is a topology, their union is in T<\/em>. Conversely, given U<\/em> \u2208<\/strong> T<\/em>, choose for each x<\/em> \u2208<\/strong> U<\/em> an element B<\/em><\/strong>x<\/em><\/sub> of B<\/em><\/strong> such that x<\/em> \u2208<\/strong> B<\/em><\/strong>x<\/em><\/sub><\/em> \u2282 U<\/em>. Then U<\/em> = \u222a<\/em>x<\/em> \u2208<\/strong> U<\/em><\/sub> B<\/em><\/strong>x<\/em><\/sub>, so U<\/em> equals a union of elements of B<\/em><\/strong>.<\/p>\n\n\n\n

      Lemma 2<\/strong>– Let X<\/em> be a topological space. Suppose that C<\/strong><\/em> is a collection of open sets of X<\/em> such that for each open set U<\/em> of X<\/em> and each x in U<\/em>, there is an element C of C<\/strong><\/em> such that x<\/em> \u2208<\/strong> C \u2282 U<\/em>. Then C<\/strong><\/em> is a basis for the topology of X<\/em>.<\/p>\n\n\n\n

      (Proof will be available on demand.)<\/p>\n\n\n\n

      Lemma 3<\/strong>– Let B<\/em><\/strong> and B<\/em><\/strong><\/em><\/strong><\/strong>‘ be bases for the topologies T<\/em> and T’<\/em>, respectively, on X<\/em>. Then the following are equivalent:<\/p>\n\n\n\n

      1. T’<\/em> is finer than T<\/em>.<\/li>
      2. For each x<\/em> \u2208<\/strong> X<\/em> and each basis for the topologies and each basis element B \u2208<\/strong> B<\/em><\/strong><\/em> containing x<\/em>, there is a basis element B’ \u2208<\/strong> B<\/em><\/strong>‘<\/em> such that x<\/em> \u2208<\/strong> B’ \u2282 B.<\/li><\/ol>\n\n\n\n

        (Proof will be available on demand.)<\/p>\n\n\n\n

        TO BE CONTINUED IN CHAPTER 2.<\/p>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"

        Chapter 1 Prerequisites- Set Theory What is topology? In simple terms, Topology is concerned with the geometric properties of objects when it undergoes physical distortions like stretching, twisting, and bending. Now we dive a bit deeper in terms of mathematics. \u00a0 Point set topology is a disease from which the human race will soon recover. Henri Poincare<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false,"footnotes":""},"class_list":["post-3075","page","type-page","status-publish","hentry"],"ams_acf":[],"yoast_head":"\nThe Topology Series - 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\/the-topology-series\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Topology Series - SOUL OF MATHEMATICS\" \/>\n<meta property=\"og:description\" content=\"Chapter 1 Prerequisites- Set Theory What is topology? 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