Homework problems

I think I messed up the axions for a closure operation in class. The fourth axiom should be that the closure of (A union B) EQUALS the union of (closure of A) and (closure of B).

I think I asserted simple containment in class.

Sufficiency of sequences: Let X be a topological space, and let x_n be a sequence of points in X. We say x_n converges to x if for every open set U containing x there exists N such that x_n is in U whenver n is larger than N.

Please think about the following assertions:

if there exists a sequence of elements of E converging to x then x is in the closure of E
the converse of this statement asserts that if x is in the closure of E then there exists a sequence of elements of E converging to x.

While the converse is not true in general it is true in many cases of interest, in particular

if the topology on is X is a metric topology then the converse holds
if the topology on is X is admits a countable basis then the converse holds