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Why topology on a set is defined the way it is?

  1. Jun 13, 2015 #1
    Following is from Wolfram Mathworld

    "A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions:

    1. The empty set is in T.

    2. X is in T.

    3. The intersection of a finite number of sets in T is also in T.

    4. The union of an arbitrary number of sets in T is also in T. " http://mathworld.wolfram.com/TopologicalSpace.html
    My question is why topology on a set is defined in this way? How these definition connect with our intuitive idea of different kind of topological space (i.e line, graphs, 3-sphere, torus etc).

    I am obviously new in topology and will be glad if you explain in basic term. I'd be specially interested to know how one differentiate between a straight line segment and a "Y" shaped graph using these definitions.

    I have convinced myself of one way, please let me know if it is correct. I can separate Y naturally in three segment let's name them a,b and c.

    Let X={a,b,c}.

    So the topology Y on X will be { {},{a,b,c},{a,b},{a,c},{b,c},{a},{b},{c}}.

    We can break a line segment on three part. Let's do likewise for line segment l.

    So the topology l on X will be { {},{a,b,c},{a,b},{b,c},{b}}. Topology Y and l are on X and obviously different. :)
     
  2. jcsd
  3. Jun 13, 2015 #2

    Svein

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    In mathematics things are defined the way they are because somebody found it a useful way of handling several (often unrelated) problem sets.
     
  4. Jun 23, 2015 #3

    Stephen Tashi

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    The definition you gave pertains to the subject called "General Topology". General Topology does not provide enough definitions and assumptions to deal with questions about the differences between a 3-sphere and a torus. To deal with differences like that you need to study "Algebraic Topology".

    General topology provides a way to treat mathematical questions involving the (vague) concept of "closeness" in a unified fashion. For example, in an elementary approach, [itex] lim_{x \rightarrow a} f(x) = L [/itex] requires one definition for real valued functions of a real variable and a different definition when x and L are two dimensional vectors. Taking the definition of "limit of a function" given by General Topology, the same definition of limit of a function applies to both cases. The only change that is made is that "open set" means one thing on the real line and a different thing in 2-dimensions.
     
  5. Jun 25, 2015 #4

    HallsofIvy

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    The whole point of topology is to generalize our concepts of "limits" and "continuity". A "topology" on a set is the collection of all open subsets of that set. And those properties are the properties of open intervals on the real line that are used in defining limits.
     
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