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More docile dfn of local path-connectedness?

  1. Apr 11, 2007 #1

    quasar987

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    I find the one on wikipedia to be a bit intimidating. It says that...

    "A topological space is said to be locally path-connected if it has a base of path-connected sets."

    We have really not covered this notion of "base" in my course, and even though I know what it is, I don't have any experience working with it. For instance, my anxieties stem from... can every topology be generated by a basis? If a topology can be generated by a basis, is it unique? If no, can a topology admit a basis of path-connected elements as well as a basis of non-path connected elements?

    So is there a friendly caracterisation of this notion? A natural candidate that immidiately comes to mind would be "A topological space X is said to be locally path-connected if every nbh of every point is itself a path-connected subspace of X."
     
    Last edited: Apr 11, 2007
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  3. Apr 11, 2007 #2

    morphism

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    The definition is just saying that for any open set you can find a path-connected open set that sits inside it.

    As for your other questions: Yes, every topology can be generated by a basis. For instance take the topology itself to be your basis! This should make you believe that bases are not unique. As a more concerete example, the conventional basis for the usual topology on the plane is the collection of open balls. At the same time, if you take the 'rectangles' whose sides are parallel to the axes you can see that they generate the usual topology as well. (To visualize this: you can fit a rectangle inside every ball, and vice versa.)

    ETA:
    Your redefinition is almost correct. It's not true that every nbhd of every point is path-connected - what is true is that every such nbhd contains a path-connected nbhd.
     
    Last edited: Apr 11, 2007
  4. Apr 11, 2007 #3

    quasar987

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    I was using the definition of nbh outlined here:http://en.wikipedia.org/wiki/Neighbourhood_(mathematics)#Definition (except that S={p}, a point)

    Are you too?
     
  5. Apr 11, 2007 #4

    morphism

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    I'm using nbhd to mean an open set - not a set that contains an open set. Although even with your definition of nbhd, your reformulation of local path-connectedness isn't correct. You are allowed to have nbhds that aren't path-connected. In fact, you're allowed to have open sets that aren't path-connected; as long as they contain an open path-connected subset, you're OK.
     
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