Can anyone explain a compact set intuitive way? I know that there are many ways to define the compactness of the set in a topological space, e.g., $X$ is compact if and only if every open cover of $X$ has a finite subcover. I also know that it is the extension of the concept of the bounded closed set in an Euclidean space to a general topological space.
However, I do not believe I truly understand what it is intuitively. Can anyone help me grasp the concept intuitively?
** I already got a wonderful answer, but please add more answers if you have different (or even similar) insight of yours. I can't examine what more I could learn on this post. I'm very excited!

Compact sets have many nice properties of finite sets but they can be infinite. For example, in finite sets:


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*All functions have a maximum

*All functions are bounded

*All sequences have a constant subsequence


In compact sets:


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*All continuous functions have a maximum

*All continuous functions are bounded

*Spaces where all subsequences have a convergent subsequence are called sequentially compact
Just like finite sets but infinite.
