Consider the set X a bp Consider j the particular point topo

Consider the set X= {a, b,p). Consider j, the particular point topology on X where p is the particular point. List the open sets in j. List the closed sets in j. Is X with j a Hausdorff space? Justify your answer.

Solution

Let X be a set. Recall that the trivial topology on X is Ttrivial = {, X} and that the discrete topology on X is Tdiscrete = {U | U X} = P(X). Clearly Ttrivial is a subset of Tdiscrete so the discrete topology on X is always finer than the trivial topology. Suppose that X contains more than one element. Let x0 be an element of X. Then the singleton set {x0} is not empty and not equal to X (since X has at least one element different than x0), and so it is not open in the trivial topology. However {x0} is open in the discrete topology (since it is a subset of X). Therefore {x0} Tdiscrete but {x0} T / trivial, which shows that the discrete topology is strictly finer than the trivial topology. Suppose that the discrete topology on X is strictly finer than the trivial topology on X. Then there is a subset U X which is open in the discrete topology but not open in the trivial topology. By the definition of Ttrivial, this means that U is nonempty and not equal to X. It follows that both U and X U are nonempty. Choose x1 U and x2 X U. Then x1 and x2 are two distinct elements of X, and this shows that X contains more than one element.