| subspace | product | image | metric | compact Hausdorff | |
| T0 | Yes | Yes | Yes | ||
| T1 | Yes | Yes | |||
| T2 | Yes | Yes | |||
| regular | Yes latex | Yes | |||
| completely regular | Yes latex | Yes | |||
| normal | Yes | ||||
| subspace | product | image | metric | compact Hausdorff | |
| connected | counterexample latex | yes | |||
| locally connected | Open | ||||
| path connected | |||||
| locally path connected | |||||
| subspace | product | image | metric | compact Hausdorff | |
| compact | counterexample latex sometimes latex | yes | Yes | ||
| locally compact | Yes latex | ||||
| limit point compact | answer answer latex | ||||
| sequentially compact | |||||
| Lindelof | yes latex | ||||
| locally compact Hausdorff | Yes latex | ||||
| subspace | product | image | metric | compact Hausdorff | |
| 2nd countable | Yes latex | ||||
| 1rst countable | |||||
| separable | not necessarily, closed yes, open no latex yes latex | ||||
| metrizable | yes | ||||
| locally metrizable |