Why do we need compactness?

As a counterexample: $\Bbb R$ is closed, and $\Bbb Z$ is an infinite subset of $\Bbb R$. However, $\Bbb Z$ has no limit points.


Your conclusion fails because you assumed that $x$, a limit point of $E$, exists.

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General Topology

Real Analysis

Compactness

Examples Counterexamples

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