In a metric space,two disjoint closed sets can be separated by two disjoint open sets.
27998
Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.
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 $d(x,A)=d(x,a_0)$
26563
Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
5323
Example to show the distance between two closed sets can be 0 even if the two sets are disjoint
133504
Suppose $X$ is a metric space and $S \subseteq X.$ Then, $S^o=\{x \in X~|~dist(x,S^c)>0\}$.
95831
Compactness and sequential compactness in metric spaces
69528
Metrizability of quotient spaces of metric spaces
88595
Prob. 21, Sec. 17 in Munkres' TOPOLOGY, 2nd ed: Closure and complementation can result in at most 14 different sets
59654
Show that in a discrete metric space, every subset is both open and closed.
91991
How do I prove that two closed sets are separable?
84791
Is every connected metric space with at least two points uncountable?
64260
Prove that a function between metric spaces is continuous iff the inverse image of a closed set is closed
2978
A and B disjoint, A compact, and B closed implies there is positive distance between both sets
13848
Showing that if a subset of a complete metric space is closed, it is also complete
52014
Showing $(C[0,1], d_1)$ is not a complete metric space
127166
Specific example of a space that is separable but not second countable.
153952
A map between metric spaces preserving convergent sequences is continuous