Re: topology questions

On 1 Sep 2005 18:03:42 -0700, "ysuzuki2" <ysuzuki2@xxxxxxxxx> wrote:

>1. What is an example of a set which is connected but not contractible?

A simple example is a 2 point space with the trivial topology, that is
X={a,b}, where the only open sets are the X and empty set. X is
connected but not contractible.

Basically, connectedness is far too general a concept to imply
contractibility. Of course contractibility implies connectedness but
it's a very specialized kind of connectedness -- a kind of
parametrization of the space by real numbers. On the other hand,
connectedness is just a lack of separation by open sets, and as a
concept, it has nothing to do with the reals.

To emphasize this point even more, it is easy to show that a
contractible space is pathwise connected, however there are lots of
examples of connected spaces which are not pathwise connected.

Ok, so a contractible space is pathwise connected, but then is the
converse true? Is every pathwise connected contractible? No again --
consider the surface of a torus.

quasi
.

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