Download PDF

closed point∗
jocaps†
2013-03-21 21:25:08
Let X be a topological space and suppose that x ∈ X. If {x} = {x} then
we say that x is a closed point. In other words, x is closed if {x} is a closed set.
For example, the real line R equipped with the usual metric topology, every
point is a closed point.
More generally, if a topological space is T1 , then every point in it is closed.
If we remove the condition of being T1 , then the property fails, as in the case
of the Sierpinski space X = {x, y}, whose open sets are ∅, X, and {x}. The
closure of {x} is X, not {x}.
∗ hClosedPointi created: h2013-03-21i by: hjocapsi version: h38512i Privacy setting: h1i
hDefinitioni h54A05i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1