Let X R For the given set determine whether or not it is cl

Let X = R For the given set determine whether or not it is closed in JU usual topology, JH half - open interval topology, JC open half - line topology, JD discrete topology, JI indiscrete topology, JF finite complement topology, or JCC countable complement topology. List only the topologies where it is closed. You do not have to justify your answer.

Solution

a) it is open interval

b)

Assume Q were open. There would be a neighborhood of 0, and so an interval containing 0 lying entirely within Q. However, each such interval contains irrational numbers, which is a contradiction.

suppose Q were closed. RQ is open. There is a neighborhood of and therefore an interval containing lying completely within RQ . however again each such interval contains rational numbers, which is a contradiction.

c) if Q is closed then RQ is open

d)

suppose Z were closed. RZ is open. There is a neighborhood of and therefore an interval containing lying completely within RZ . however again each such interval contains rational numbers, which is a contradiction.

if Z is closed then RZ is open

e)closed interval

f) it is open

g) closed interval

f) it is open interval