example for a non compact space?

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April 3rd 2011, 01:52 AM
aharonidan

example for a non compact space?

hi
I need to find examples for these two topological spaces:
a. A space $X$ such that every sequence has a convergent subsequence, but X is not compact.
b. Space $X$ . every infinite set $S \subseteq X$ has an accumulation point and $X$ is not compact.

thanks
April 3rd 2011, 07:03 AM
roninpro

You probably need to look at topological spaces that are not metrizable. If $X$ were a metric space, then condition (a) would actually be equivalent to compactness.
April 3rd 2011, 05:08 PM
Drexel28

Quote:

Originally Posted by aharonidan

b. Space $X$ . every infinite set $S \subseteq X$ has an accumulation point and $X$ is not compact.

Consider $X=[0,1]^\Omega$ where $\#(\Omega)>\aleph_0$ with the product topology and define

$\mathcal{X}=\left\{(x_\omega)\in X:x_\omega=0\text{ for all but countably many }\omega\in\Omega\right\}$

Let $Y$ be an infinite subset of $\mathcal{X}$ . That space does the trick. Try proving it.

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