Hausdorff metric and connectedness
133938
Connectedness of the Hausdorff distance.
149980
Are there any countable Hausdorff connected spaces?
154682
G/H is Hausdorff implies H is closed (General topology, Volume 1 by N. Bourbaki)
123742
Proving connectedness of the $n$-sphere
7402
Showing a compact metric space has a countable dense subset
27998
Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.
90628
The metric spaces are Tychonoff spaces
96781
connected manifolds are path connected
42811
Product of path connected spaces is path connected
151719
Connected set in $\mathbb{S}^1$
40208
Locally Constant Functions on Connected Spaces are Constant
107701
Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$
146967
Understanding equivalent metric spaces
39763
Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure
90110
Hyperbolic metric geodesically complete
93986
Connectivity, Path Connectivity and Differentiability
11455
Connected metric spaces with at least 2 points are uncountable.