complete metric space $X$ and nested sequence of closed sets $A_m \subset X$ where $\bigcap_{n=1}^\infty = \emptyset$
128844
An empty intersection of decreasing sequence of closed sets
154978
Some equivalent formulations of compactness of a metric space
86970
Let $\{x_n\}$ be a sequence in a metric space such that $\{x_{2n}\}$, $\{x_{2n+1}\}$ and $\{x_{3n}\}$ all convergent. Show that $\{x_{n}\}$ converges
52014
Showing $(C[0,1], d_1)$ is not a complete metric space
154796
Completeness of sequence of reals with finitely many nonzero terms
73080
Is the set of all bounded sequences complete?
133152
(X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that ∀𝑥 ∈ X, B(x,ε) is contained in some member of the cover.
17842
Is every set in a separable metric space the union of a perfect set and a set that is at most countable?
10127
Example of a sequence of closed and connected sets whose intersection is disconnected
44673
$M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time
135315
Proving whether a function on the set of real sequences is a metric
17067
To prove that $(\mathbb{P(N)},d)$ is a metric space
140692
Existence of a complete metric on $(0,1)$
90628
The metric spaces are Tychonoff spaces
95831
Compactness and sequential compactness in metric spaces
74877
Every metric space with a countable base is separable
109638
Show that a complete metric space without isolated points is uncountable