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Topology, MM8002/SF2721, Spring 2015.
Separation axioms
Let (X, T ) be a topological space. For a point x ∈ X, let Tx denote the set of
open neighborhoods of x in the topology T . In other words,
Tx = {U ∈ T | x ∈ U } .
Similarly, for a subset A ⊆ X, let
TA = {U ∈ T | A ⊆ U } .
Separation axioms:
T0 : If x 6= y, then Tx 6= Ty .
T1 : If x 6= y, then Tx and Ty are incomparable, i.e., Tx 6⊆ Ty and Ty 6⊆ Tx .
T2 : (Hausdorff) If x 6= y, then there are U ∈ Tx and V ∈ Ty such that U ∩V = ∅.
T3 : (Regular) T1 holds and for every point x ∈ X and every closed subset
A ⊆ X such that x 6∈ A, there are U ∈ Tx and V ∈ TA such that U ∩ V = ∅.
T4 : (Normal) T1 holds and if A and B are disjoint closed subsets of X, then
there are U ∈ TA and V ∈ TB such that U ∩ V = ∅.
Some facts:
• We have T4 ⇒ T3 ⇒ T2 ⇒ T1 ⇒ T0 . All implications are strict.
• The indiscrete topology T = {∅, X} on a set X with > 1 elements does not
satisfy T0 .
• The topological space (X, T ) = ({a, b}, {∅, {a}, {a, b}}) satisfies T0 but not
T1 .
• The finite complement topology on an infinite set satisfies T1 but not T2 .
• Every metric space is normal.
• Every normal space with a countable base is metrizable (this is a deep
result called the Urysohn metrization theorem. We will not prove it in this
course. A proof can be found in Munkres).
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