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Topology

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Open, Closed, Compact in Complete Metric Space

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Let $X$ be a complete metric space, and let $A\subset B \subset X$, where $A$ is an open set, and $B$ is a closed set.

Recall that $K$ is a compact subset of $X$ if for every sequence $(x_n)_n$ in $K$ there exists a subsequence $(x_{n_k})_{k\geq 1}$ which converges to a point in $K$.

Pick the correct statement from the list:

A

$B$ must be a compact set.

B

$A$ cannot equal $B$.

C

it must be that ${\rm Cl}(A)\subset {\rm Int}(B)$, where ${\rm Cl}(A)$ is the closure of $A$, and $ {\rm Int}(B)$ is the interior of $B$.

D

every Cauchy sequence $(x_n)_n$ contained in $B$ is necessarily converging to a point $x_0\in B$.