Difficult# Topological Equivalence, Convergence, Completeness

TOPO-YZ3EVB

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**: