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Topology

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Topological Equivalence, Convergence, Completeness

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Let $(X,d_1)$ and $(X,d_2)$ be two metric spaces.

Recall that if $U\subset X$ is not open in $(X,d)$, then there exists some $z\in U$ such that for every $\epsilon>0$ one can find a point $y_\epsilon\not\in U$ inside the open ball $B(z,\epsilon):=\{u\in U: d(z,u)<\epsilon\}$.

Choose ALL the statements which are MUTUALLY EQUIVALENT:

A

A sequence $(x_n)_n$ is converging in $(X,d_1)$ if and only if it is converging in $(X,d_2)$.

B

A sequence $(x_n)_n$ is Cauchy in $(X,d_1)$ if it is Cauchy in $(X,d_2)$.

C

$(X,d_1)$ and $(X,d_2)$ are topologically equivalent.

D

$B\subset X$ is closed in $(X,d_1)$ if and only if it is closed in $(X,d_1)$.

E

$A\subset X$ is open in $(X,d_1)$ if and only if it is open in $(X,d_1)$.

F

$(X,d_1)$ is complete if and only if $(X,d_2)$ is complete.