Download Sierpinski space

Sierpinski space∗
CWoo†
2013-03-21 13:03:59
Sierpinski space is the topological space X = {x, y} with the topology given
by {X, {x}, ∅}.
Sierpinski space is T0 but not T1 . It is T0 because {x} is the open set
containing x but not y. It is not T1 because every open set U containing y
(namely X) contains x (in other words, there is no open set containing y but
not containing x).
Remark. From the Sierpinski space, one can construct many non-T1 T0
spaces, simply by taking any set X with at least two elements, and take any nonempty proper subset U ⊂ X, and set the topology T on X by T = P (U ) ∪ {X}.
∗ hSierpinskiSpacei created: h2013-03-21i by: hCWooi version: h31222i Privacy setting:
h1i hDefinitioni h54G20i
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