- #1
complexnumber
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If (X,\tau) is either a T_1 space or Hausdorff space then for any x \in X the singleton set \{ x \} is closed.
Why is this the case? I can't see the reason from the definitions of the spaces.
Definition:
Let (X,\tau) be a topological space and let x,y \in X be any two
distinct points, if there exists any two open sets A,B \in \tau
such that x \in A but x \notin B and y \in B but y \notin A,
then (X, \tau) is a T_1 space.
Definition:
A topological space (X, \tau) is Hausdorff if for any
x,y \in X, x \ne y, \exists \text{ neighborhoods } U \ni x and
V \ni y such that U \cap V = \varnothing.
Why is this the case? I can't see the reason from the definitions of the spaces.
Definition:
Let (X,\tau) be a topological space and let x,y \in X be any two
distinct points, if there exists any two open sets A,B \in \tau
such that x \in A but x \notin B and y \in B but y \notin A,
then (X, \tau) is a T_1 space.
Definition:
A topological space (X, \tau) is Hausdorff if for any
x,y \in X, x \ne y, \exists \text{ neighborhoods } U \ni x and
V \ni y such that U \cap V = \varnothing.