What are the examples of open sets that are compact?

Is the entire space X compact?

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- Oct 25th 2009, 10:54 PMfelixmcgradycompactness
What are the examples of open sets that are compact?

Is the entire space X compact? - Oct 26th 2009, 11:26 AMaliceinwonderland
It depends on what topology you give on the space X.

For example, if you give a discrete topology on a finite set X, then every open set is compact.

Another example is that every open set is compact in the finite complement topology on $\displaystyle \mathbb{Re}$.

Meanwhile, if you give a usual topology on $\displaystyle \mathbb{Re}$, then an open compact set is an empty set only.

If you give a subspace topology on A with respect to the usual topology on $\displaystyle \mathbb{Re}$, where A is a finite subset of $\displaystyle \mathbb{Re}$, then every open set in the topological space A is compact.