Show that every subset of a discretespace is both open and closed.

Solution Preview

Proof:
First, let's clarify the definition of a discrete space.
Given a set X, the discrete topology on X is defined by ...

Solution Summary

Open and closed discrete spaces are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Let X and Y connected, locally path connected and Hausdorff. let X be compact.
Let f: X ---> Y be a local homeomorphism. Prove that f is a surjective covering with finite
fibers.
Prove:
a) Any subspace of a weak Hausdorff space is weak Hausdorff.
b)Any open subset U of a compactly generated space X is compactly generated

Which of the following topological spaces is normal?
a) Reals with the "usual topology."
b) Reals with the "finite complement topology:" U open in X if U - X is finite or is all of X.
c) Reals with the "countable complement topology:" U open in X if X - U is countable or is all of X.
d) Reals with the "lower limit topolog

Please help with the following problem.
For the following, I'm trying to decide (with proof) if A is a closed subset of Y with respect to the topology, T
(i) Y = N, T is the finite complement topology, A = {n e N | n^2 - 2011n+1 < 0}.
(ii) Y = R, T is the usual topology, A is the set of irrational numbers between 0 and

(See attached file for full problem description)
1. Show that the functions d defined below satisfy the properties of a metric.
a. Let X be any nonempty set and let d be defined by The d is the call the discrete metric.
b. If X is the set of all m-tuples of real numbers and, if for and , then (X,d) is a metric spac

Please solve the problems in B.pdf file by using the textbook.
The topics covered in this problem set are:
1. Metric spaces. Basic concepts ( Sections 5.1-5.2 )
Problems: - Section 5--# 1,2,4,5,6,7
2. Convergence. Openandclosed sets (Sections 6.1-6.6)
Problems: - Section 6--# 1,2,3,4,5,9,10
3. Complete m

Would you agree with my answers to the following questions?
1. "What is it like for you when you get depressed?" OPEN ENDED
2. "Have you had a physical exam in the last two years?" CLOSED
3. "Are you saying you don't give up easily?" CLOSED
4. "How does this job affect your mood

Prove:
If H and K are disjoint closed point sets, then there exist open point sets U and V containing H and K respectively such that cl(U) and cl(V) are disjoint.

Let P : C -> R be defined by P(z) = Re z; show that P is an open map but it is not a closed map. ( Hint: Consider the set F = { z : Imz = ( Re z)^-1 and Re z doesn't equal to 0}.) Please explain every step and justify.