Thread: example for a non compact space?

1. example for a non compact space?

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

thanks

2. You probably need to look at topological spaces that are not metrizable. If $\displaystyle X$ were a metric space, then condition (a) would actually be equivalent to compactness.

3. Originally Posted by aharonidan b. Space $\displaystyle X$. every infinite set $\displaystyle S \subseteq X$has an accumulation point and $\displaystyle X$ is not compact.
Consider $\displaystyle X=[0,1]^\Omega$ where $\displaystyle \#(\Omega)>\aleph_0$ with the product topology and define

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

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

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