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# SEPARATION AXIOMS | Topology

S
Shruti K.
26/09/2018 0 0

Regular Spaces:The properties of topological spaces are in general quite different from those of metric spaces, so some additional restrictions are often imposed on a topology in order to bring the properties of the corresponding
space closer to those of metric spaces.

Definition: A space X is T3 if for each point x ∈ X and each closed set F ⊆ X with x /∈ F, there are disjoint open sets U and V such that x ∈ U and F ⊆ V . The space X is called regular if it satisfies both T1 and T3 axioms.

Normal Spaces: Topological spaces satisfying this condition are quite close to metric spaces; in fact, it gives a characterisation of second countable metric spaces. We will also see that a continuous function defined on a closed subspace of a topological space with this property admits a continuous extension to the whole space.

Definition: A space X is T4 if each pair of disjoint closed subsets of X have disjoint nbds. X is normal if it satisfies both the T1 and T4 axioms.

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