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Topology

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Topological Equivalence, Convergence, Compactness

TOPO-FLHVZK

Let $(X,d)$ be a metric space. A set $K\subset X$ is called (sequentially) relatively compact if every sequence $(x_n)_n$ in $X$ entirely contained in $K$ has a converging subsequence.

If, in addition, all such subsequential limits are also contained in $K$, then $K$ is said to be a (sequentially) compact subset of $X$.

Choose ALL the statements from the list which must be true:

A

If $K$ is relatively compact and closed, then $K$ is compact.

B

if $K$ is relatively compact and open, then the complement $K^c$ is compact.

C

if $(X,d_1)$ is a topologically equivalent metric space to $(X,d)$, then $K$ is compact in $(X,d_1)$ if and only if it is compact in $(X,d)$.

D

if $(X,d_1)$ is a topologically equivalent metric space to $(X,d)$, then $K$ is relatively compact in $(X,d)$ if and only if it is relatively compact in $(X,d_1)$.

E

If $B\subset K\subset X$ and $B$ is closed while $K$ is relatively compact, then $B$ is compact.

F

If $A\subset K\subset X$ and $K$ is relatively compact, then $A$ is also relatively compact.