every second countable space is separable
Let be a second countable space and let be a countable base. For every non-empty set in , choose a point . The set of all such points is clearly countable and it’s also dense since any open set intersects it and thus the whole space is the closure of . That is, is a countably dense subset of . Therefore, is separable. ∎
- 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
|Title||every second countable space is separable|
|Date of creation||2013-03-22 12:22:10|
|Last modified on||2013-03-22 12:22:10|
|Last modified by||drini (3)|