Let X R For the given set determine whether or not the set

Let X = R For the given set determine whether or not the set is open in TU usual topology, TH half - open interval topology, TC open half - line topology, TD discrete topology, TI indiscrete topology, TF finite complement topology, or TCC countable complement topology. List only the topologies where it is closed. You do not have to justify your answer.

(a) (1, 5]

(b) Q

(c) R Q

(d) R Z

(e) {1/n : n Z+}

(f) R {1/n : n Z+}

(g) {1/n : n Z+} {0}

(h) R – ({1/n : n Z+} {0})

Solution

Let us write N for \"neither open nor closed\". Also, it is assumed that TH denotes the left half open topology; if your version is right half open, you need to change the answer suitably. The following table provides the answer: