{"id":2877,"date":"2014-02-24T09:44:56","date_gmt":"2014-02-24T14:44:56","guid":{"rendered":"http:\/\/www.goodmath.org\/blog\/?p=2877"},"modified":"2014-02-24T09:44:21","modified_gmt":"2014-02-24T14:44:21","slug":"topological-spaces-defining-shapes-by-closeness","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2014\/02\/24\/topological-spaces-defining-shapes-by-closeness\/","title":{"rendered":"Topological Spaces: Defining Shapes by Closeness"},"content":{"rendered":"

When people talk about what a topological space is, you’ll constantly hear one refrain: it’s just a set with structure!<\/p>\n

I’m not a fan of that saying. It’s true, but it just doesn’t feel right to me. What makes a set into a topological space is a relationship between its members. That relationship – closeness<\/em> – is defined by a structure in the set, but the structure isn’t the point; it’s just a building block that allows us to define the closeness relations.<\/p>\n

The way that you define a topological space formally is:<\/p>\n

A topological space<\/em> is a pair (X, T, N), where X is a set of objects, called points<\/em>; T is a set of subsets of X; and N is a function from elements of X to elements of T (called the neighborhoods<\/em> of X where the following conditions hold:<\/p>\n

    \n
  1. Neighborhoods basis: \\forall A \\in N(p): p \\in A: every neighborhood of a point must include that point.<\/li>\n
  2. Neigborhood supersets: \\forall A \\in N(p): \\forall B \\in X: B \\supset A \\Rightarrow B \\in N(p). If B is a superset of a neighborhood of a point, then B must also be a neighborhood of that point.<\/li>\n
  3. Neighborhood intersections: \\forall A, B \\in N(p): A \\cap B \\in N(p): the intersection of any two neighborhoods of a point is a neighborhood of that point.<\/li>\n
  4. Neighborhood relations: \\forall A \\in N(x): \\exists B \\in N(x): \\forall b \\in B: A \\in N(b). If A is a neighborhood of a point p, then there’s another neighborhood B of p, where A is also a neighborhood of every point in B.<\/li>\n<\/ol>\n

    The collection of sets T is called a topology<\/em> on T, and the neighborhood relation is called a neighborhood topology<\/em> of T.<\/p>\n

    Like many formal definitions, this is both very precise, and not particularly informative. What the heck does it mean?<\/em><\/p>\n

    In the previous topology post, I talked about metric spaces<\/a>. Every metric space is<\/em> a topological space (but not vice-versa), and we can use that to help explain how the set-of-sets T defines a meaningful definition of closeness for a topological space.<\/p>\n

    In the metric space, we define open balls<\/em> around each point in the space. Each one forms an open set<\/em> around the point. For any point p in the metric space, there are a sequence of ever-larger open-balls of points around p. <\/p>\n

    That sequence of open balls defines the closeness relation in the metric space: <\/p>\n