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Theorem:  A space X is a T1-space if and only if each finite subset of X is closed.

Week 5

Research the separation axioms distinguishing T0, T1, T2, T3, and T4 spaces. Discuss some interesting aspects of these spaces to include their subtlety. Note that it is easy to create a space that is not even T0, e.g. X = {a,b,c,d,e} with Ƭ = {∅, X, {a,b,d},{a,b}, {d}} where the points a ∈ X and b ∈ X cannot be separated. As always comment on your classmates’ observations about the separation axioms, and remember to cite your work.
For example: A theorem regarding T1 spaces:
Theorem:  A space X is a T1-space if and only if each finite subset of X is closed.

[Recall that X is T1 if, for any pair (x,y) of distinct elements of X, there are open sets U and V such that U contains x but not y and V contains y but not x.]

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Proof:  Suppose each finite subset of X is closed and consider distinct points a,b ∈ X.  Then U = X‒{a} and U = X‒{b} are open sets since {a} and {b} are closed.  Moreover, U contains b but not a, and V contains a but not b; thus X is T1.  Now suppose X is T1.  It is sufficient to prove that each singleton set {a} is closed since any finite set is the union of a finite number of singleton sets.  To show {a} is closed, we show U = X‒{a} is open.  For all b ∈ X where b ≠ a there is an open set Ub containing b but not a since X is T1.  The union of the collection of sets Ub for all b ∈ X is open by the definition of a topology, but the union of all sets Ub is X‒{a}.

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