Vipul: Created page with "{{topospace property implication| stronger = ultraconnected space| weaker = normal space}} ==Statement== Any ultraconnected space is a normal space. ==Definitions u..."

2012-01-28T17:54:14Z

Created page with "{{topospace property implication| stronger = ultraconnected space| weaker = normal space}} ==Statement== Any ultraconnected space is a normal space. ==Definitions u..."

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{{topospace property implication|
stronger = ultraconnected space|
weaker = normal space}}

==Statement==

Any [[ultraconnected space]] is a [[normal space]].

==Definitions used==

{| class="sortable" border="1"
! Term !! Definitions used
|-
| [[ultraconnected space]] || A space <math>X</matH> is ultraconnected if any two non-empty closed subsets have non-empty intersection.
|-
| [[normal space]] || A space <math>X</math> is normal if, given any two disjoint closed subsets <math>A,B</matH> of <math>X</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</matH> such that <math>A \subseteq U, B \subseteq V</math>.
|}

==Proof==

The proof is immediate from the observation that in an ultraconnected space, it is not possible to have disjoint non-empty closed subsets. Hence, given disjoint closed subsets, one of them must be empty, and we can use the empty space and whole space as the corresponding open subsets.

Ultraconnected implies normal - Revision history

Vipul: Created page with "{{topospace property implication| stronger = ultraconnected space| weaker = normal space}} ==Statement== Any ultraconnected space is a normal space. ==Definitions u..."