Every subspace of a separable metric space is separable.
19856
Prove that a subset of a separable set is itself separable
31686
Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set
16892
Reference request: product Borel $\sigma$-algebra of non-separable metric spaces
35652
Every compact metric space is complete
119649
Every metrizable space with a countable dense subset has a countable basis
154724
Every vector space has a basis
118508
Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact?
10034
Give an example of a non-separable subspace of a separable space
127522
Space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable
124179
Proof that every metric space is normal
62034
If every open set of a metric space $(X, \rho)$ is $\mu^*$-measurable, then $\mu^*$ is a metric outer measure.
50946
Separable Hilbert space have a countable orthonormal basis
88185
Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed
27517
Isometry in compact metric spaces
125580
Proving that every countable metric space is disconnected?
109991
Every convergent sequence in a metric space has a unique limit.
151984
Proving the existence of a metric space