proving that the subspace topology and the metric topology are the same
137438
Restrict a metric, gives same topology as subspace topology from larger space $X$
108619
Proving Baire's theorem: The intersection of a sequence of dense open subsets of a complete metric space is nonempty
77563
Homeomorphism between spaces equipped with cofinite topologies
128730
Open subspaces of locally compact Hausdorff spaces are locally compact
85242
Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$
107701
Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$
27998
Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.
39763
Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure
126944
Metric spaces problem
104881
Showing that two Banach spaces are homeomorphic when their dimensions are equal.
6415
Distance function is continuous in topology induced by the metric
49563
Cauchy nets in a metric space
62479
A isometric map in metric space is surjective?
132641
Proving that Euclidean Space is a complete metric space
146485
Prove that the product space is a metric space.
67191
When is the union of a family of subspaces of a vector space also a subspace?
66239
Verifying that a certain collection of intervals of $\mathbb R$ forms a topology