Recent content by zilla

If X is a T1, 1st countable topological space and x is a limit point of A in X, then there exists a sequence {bn} in A whose limit is x. (I'm doing this class through independent study, and in this last session the prof decided we hadn't covered enough in the semester (even though we've...

Thanks - I was pretty sure, but wanted to check.

Homework Statement Give an example of two metric spaces (X1, d1) and (X2, d2) which are topologically equivalent and for which (X1, d1) is complete and (X2, d2) is not. 2. The attempt at a solution The open unit disc and R2. They are homeomorphic, but there are Cauchy sequences in the...

Thanks for the help - I appreciate it.

Could you elaborate on what you mean by the 'first property'? Also, so I'm looking for |f(x) - f(y)| < Epsilon?

Homework Statement Let (X,d) be a metric space, M a positive number, and f: X->X a continuous function for which: d(f(x), f(y)) is less than or equal to Md(x,y) for all x, y in X. Prove that f is continuous. Use this to conclude that every contractive function is continuous. The...

Homework Statement Suppose that each of A and B is a set such that: (a) A is a subset of [0,1], B is a subset of [0,1] (b) Neither of A or B is empty (c) 0 (zero) is an element of A (d) The union of A & B = [0,1] (e) A & B are disjoint (f) A contains no limit points of B...