Week 5 (418)

Week 5Research 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.]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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