$M$ is metric space. $A\subset M, F$ is closed in $M$, prove $A\cap F$ is closed in $A$
86317
Show that $F\subset Y$ is closed in $Y$ iff $F=Y\;\cap\;H$ where $H\subset X$ is closed in $X$.
124179
Proof that every metric space is normal
18040
Show that a compact metric space is bounded
15480
Open set in a metric space is union of closed sets
148994
$K \subset \mathbb{R}^n$ is compact iff it is closed and bounded
43248
Closure of a Ball is Contained in Closed Ball (usual metric)
114608
Proof concerning regular space: there exist a closed set contained in any open set containing $x$
16735
If $f,g:X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\,|\,f(x)=g(x)\}$ is closed
24050
Characterization of the Closure of a Set
59654
Show that in a discrete metric space, every subset is both open and closed.
88185
Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed
112303
Distances between closed sets on metric spaces
73368
Does this proof that every open set in a metric space is a union of open balls use the axiom of choice?
16060
Showing this metric space is complete
151984
Proving the existence of a metric space
110744
How to prove that a compact set in a Hausdorff topological space is closed?
106434
Prove that for a dense set $A$ and an open set $U$, $U\subset \overline{A\cap U}$