closed set in a subspace
In the following, let be a topological space.
If is closed in , then is open (http://planetmath.org/OpenSet) in , and by the definition of the subspace topology, for some open . Using properties of the set difference (http://planetmath.org/SetDifference), we obtain
On the other hand, if for some closed , then , and so is open in , and therefore is closed in . ∎
Suppose is a topological space, is a closed set equipped with the subspace topology, and is closed in . Then is closed in .
This follows from the previous theorem: since is closed in , we have for some closed set , and is closed in . ∎
|Title||closed set in a subspace|
|Date of creation||2013-03-22 15:33:32|
|Last modified on||2013-03-22 15:33:32|
|Last modified by||yark (2760)|