Disjoint closed sets in a metric space
5323
Example to show the distance between two closed sets can be 0 even if the two sets are disjoint
3896
A closed set in $(A, \mathcal T_A)$ that is not closed in $(X, \mathcal T)$
144353
A closed ball in a metric space is closed
84791
Is every connected metric space with at least two points uncountable?
11383
Disjoint compact sets in a Hausdorff space can be separated
132641
Proving that Euclidean Space is a complete metric space
49721
Prove the empty set is closed for a metric space (X,d).
119105
If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$
82628
Cardinality of a set of closed intervals
123090
How to show that disjoint closed sets have disjoint open supersets?
30231
The boundary of union is the union of boundaries when the sets have disjoint closures
44623
Closed set as a countable intersection of open sets
11455
Connected metric spaces with at least 2 points are uncountable.
133504
Suppose $X$ is a metric space and $S \subseteq X.$ Then, $S^o=\{x \in X~|~dist(x,S^c)>0\}$.
144731
Disjoint closed sets in a second countable zero-dimensional space can be separated by a clopen set
60868
Interplay of Hausdorff metric and Lebesgue measure
151984
Proving the existence of a metric space