How to proof an open set and its complement contained in a metric space with infinite element are both infinite?
98553
Infinite metric space has open set $U$ which is infinite and its complement is infinite
157753
Show that any set in a metric space can be written as the intersection of open sets
112303
Distances between closed sets on metric spaces
52014
Showing $(C[0,1], d_1)$ is not a complete metric space
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.
111008
Show that $X $ is a infinite set ,it is connected in the finite complement topology?
19289
complement of a set and its measure
35573
A closed ball in a metric space is a closed set
151295
Proving $|S \times \mathbb{N}|=|S|$ when $S$ is an infinite set
55614
Prove that a set consisting of a sequence and its limit point is closed
118508
Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact?
99236
Interior of closure of an open set
36281
A set is infinite iff it is equivalent to a proper subset of itself
116009
Proving that if a set is both open and closed then it is equal to the real numbers
134071
If A is a finite set and B is an infinite set, then B\A is an infinite set
124179
Proof that every metric space is normal
7402
Showing a compact metric space has a countable dense subset