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Relations between compactness and connectedness

  1. Oct 10, 2012 #1
    Hello there,

    This might be probably a simple question, but my wondering was:

    Is there any relation between the compactness and the connectedness of a topological space?

    Let us consider the specific example (of interest for me) of a subdomain D of a 3D Riemannian manifold.

    i) If D is compact, can I say that it is necessarily connected?

    ii) If D is simply-connected, can I say that it is compact?

    iii) If D is multi-connected, can I say that it is compact?

    If one or another answer to these questions is negative, can you please provide me with an example?

    Regards
     
  2. jcsd
  3. Oct 10, 2012 #2

    HallsofIvy

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    Let D be the ball of radius 1 with center at (-1, 0, 0) union the ball with radius 1 and center at (1, 0, 0) in R3. That set is compact but not connected.

    The line, R1 is simply connected but not compact.

    I don't know what you mean by "multi- connected"- has multiple components?
    If so, the union of [itex][1, \infty)[/itex] and [itex](-\infty, -1][/itex] has two components but is not compact. The set of all integers, as a subset of the real line, has an infinite number of components but is not compact.

    The answer to all of those questions is negative. I really have no idea why you would think that "connected" and "compact" are related. They both start with "c" is about as close as you will get!
     
  4. Oct 10, 2012 #3
    Thanks a lot for your answer and your examples HallsofIvy!
     
  5. Oct 10, 2012 #4

    mathwonk

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    i'm going to go out on a limb here and maybe say something false, in an attempt to show that the three concepts are quite different.

    1) a manifold M is connected if every continuous map from {0,1}-->M extends to a continuous map [0,1]-->M.

    2) a manifold M is simply connected if every continuous map circle-->M

    extends to a continuous map disc-->M.

    3) a manifold M is compact if there is a surjective continuous map [0,1]-->M.

    so you see these three properties are pretty dissimilar.
     
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