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.
Solution Summary
This solutions contains a proof regarding disjoint closed point sets. It is solved in an attached Word document.
Let A and B be subsets of R show that if there exists disjoint open sets U and V with A subset or equal of U and B subset or equal of V then A and B are separated.
Please see the attached file for the fully formatted problems.
B6. (a) Define what it means for a topological space to be connected.
(b) Suppose that A and B are subspaces of a topological space X, and that U C A fl B is open in both A and B in the relative topologies. Show that U is open in A U B in the relative topology.
2. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X?
3. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.
2.) If " S " is the set of all "x" such that 0≤x≤1, what points, if any, are points of accumulation of both "S" and C(S)?
3.) Prove that any finite set is closed.
5.) Prove that, if "S" is open, each of its points is a point of accumulation of "S".
1.) Suppose "S" is a set having the number "M" as its least up
Please help with the following problem.
Use the following steps to prove that every non-empty open subset of R is a union of at most countably many disjoint open intervals.
Suppose that G is a non-empty open subset of R.
1. For each a <- G let Ia be the union of all those open intervals I which contain a and are containe
I have these problems from Topology of Surfaces by L.Christine Kinsey: the problems I require assistance with are 2.26, 2.28, 2.29, and 2.32. These are stated below.
PROBLEM (Exercise 2.26). Describe what stereographic projection does to
(1) the equator,
(2) a longitudinal line through the north and south poles,
(3) a tr
