If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0$ always?
5323
Example to show the distance between two closed sets can be 0 even if the two sets are disjoint
42266
Show that $d(x,y)=\min \{1,|x-y|\}$ is a metric on $\mathbb R$
17067
To prove that $(\mathbb{P(N)},d)$ is a metric space
5251
If $F$ is closed, does there exist a finite measure on $[0,1]$ with support $F$?
127972
Can distance between two closed sets be zero?
2056
A space is $T_{0}$ if and only if the closures of singletons are distinct
11383
Disjoint compact sets in a Hausdorff space can be separated
127919
prove that if $E$ is connected and $E \subseteq F \subseteq \overline{E}$, then $F$ is connected.
123753
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
94396
A topological space is extremally disconnected iff every two disjoint open sets have disjoint closures
112323
Showing that $f = 0 $ a.e. if for any measurable set $E$, $\int_E f = 0$
154978
Some equivalent formulations of compactness of a metric space
158155
Is set of isolated points of S closed?
26215
Proof Verification: Let $D=\{ \frac{1}{n} \mid n\in\mathbb{N} \} \cup\{0\} \subset \mathbb{E}^1$. Prove that $D$ is compact.
144353
A closed ball in a metric space is closed
79011
If $(F:E)<\infty$, is it always true that $\operatorname{Aut}(F/E)\leq(F:E)?$
121135
Why don't we use closed covers to define compactness of metric space?