{"id":127,"date":"2016-09-09T13:18:49","date_gmt":"2016-09-09T17:18:49","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=127"},"modified":"2017-08-22T17:01:50","modified_gmt":"2017-08-22T21:01:50","slug":"an-introduction-to-topological-spaces","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2016\/09\/09\/an-introduction-to-topological-spaces\/","title":{"rendered":"An Introduction to Topological Spaces"},"content":{"rendered":"

In class last week, we learned about the uniqueness of limits. Specifically, the limit of a sequence, when it exists, must be unique. But what if I told you that there existed a place in mathematics where this is not always the case?\u00a0Today, I\u2019d like to briefly define that context\u00a0and then illustrate some basic examples. Finally, I will note how our new and old mathematical analysis knowledge\u00a0converge<\/em>\u00a0(forgive the pun) when we look at sequences from the perspective of this amazing field of mathematics\u00a0called a topology.<\/p>\n

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A topological space \"(X, consists of a set \"X\" and a topology \"\tau\", which is a collection of subsets of \"X\" called \"\tau\". For \"\tau\" to be a topology, it must satisfy three requirements:<\/p>\n

    \n
  1. \"\emptyset,<\/li>\n
  2. Given any collection of sets in \"\tau\", \"\tau\" must contain their union.<\/li>\n
  3. Given any finite collection of sets in \"\tau\", \"\tau\" must contain their intersection.<\/li>\n<\/ol>\n

    Let\u2019s begin by looking at the trivial topology on \"\mathbf{R}\" where \"X=\mathbf{R} Does this \"\tau\" satisfy the requirements of a topology? Let\u2019s check:<\/p>\n

      \n
    1. Trivial<\/li>\n
    2. \"\tau\" contains the unions of all of the collection of sets in \"\tau\":\n