My problem is that the space [itex] X= (0,1)[/itex] is not sequentially compact and compact at the same time.(adsbygoogle = window.adsbygoogle || []).push({});

It is not sequentially compact:

If we define the sequence [itex](\frac{1}{n}) [/itex] we can show that it is not sequentially compact as the sequence converges to 0, but [itex] 0 \notin X[/itex].

It is compact:

On the other hand, for X to be compact we need

1) bounded: The space X is bounded as any ball with center [itex] x \in X [/itex] and radius 2 will X.

2) closed: Is closed as its complement is the empty set (which is open)

Thus, the set [itex] X [/itex] is compact, which is a contradiction as X is not sequentially compact.

Where is my mistake when I show that X is compact?

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# I Why the space X=(0,1) is (not sequentially) compact?

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