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Possible Duplicate:
Compactness is closed-hereditary - significance of closed property?

Proposition let $(M,d)$ be a metric space where $K\subset M$ is compact and $A\subset K$ is closed. Then $A$ is compact.

Proof let $\{ G_\alpha \}_{\alpha\in I}$ be a open cover of $A$, then $ A \subset \cup_{\alpha\in I} G_\alpha $. Since $A^c$ is open and $K\setminus A \subset A^c$, $$ K \subset \left( \bigcup_{\alpha\in I} G_\alpha \cup A^c\right) \quad (1) $$ and, since $K$ is compact, $\exists N\in\mathbb{N}$, $\exists \alpha_1,...,\alpha_N \in I$: $$ A\subset K \subset \left( \bigcup_{i\in I}^N G_{\alpha_i} \cup A^c\right) \quad (2) $$ Then, $A$ is compact.

Question Why $A$ is necessarily closed? Step (1) is also true if $A$ is open since $\cup_{\alpha\in I} G_\alpha$ covers $A$, and $A\cup A^c = M$.

Thanks in advance.

  • 4
    "Since $A^c$ is open..."2012-12-07
  • 0
    I am fairly sure I answered this sort of question not long ago.2012-12-07
  • 0
    Yes, I answered [this question](http://math.stackexchange.com/questions/243745/compactness-is-closed-hereditary-significance-of-closed-property/).2012-12-07

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