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Comprehensive Examination - Topology: A A/Bmeans (A:a Aanda/ B)

This document provides a comprehensive examination in topology for Spring 2004. It consists of 7 problems related to topics in topology, such as continuity of functions between topological spaces, properties of metric spaces, product topologies, convergence of sequences, and properties of the lower limit topology on the real line. Students are instructed to complete any 5 of the 7 problems, each worth 20 points.

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Süleyman Cengizci
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239 views1 page

Comprehensive Examination - Topology: A A/Bmeans (A:a Aanda/ B)

This document provides a comprehensive examination in topology for Spring 2004. It consists of 7 problems related to topics in topology, such as continuity of functions between topological spaces, properties of metric spaces, product topologies, convergence of sequences, and properties of the lower limit topology on the real line. Students are instructed to complete any 5 of the 7 problems, each worth 20 points.

Uploaded by

Süleyman Cengizci
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Department of Mathematics

California State University, Los Angeles

Comprehensive Examination Topology


Spring 2004
Beer*, Chabot, Verona
Do any five of the problems that follow. Each problem is worth 20 points. The set of positive
integers is denoted by N , the set of rationals by Q, and the set of real numbers by R. The notation
Ac means the complement of the set A with respect to an understood universal set. The notation
A \ B means {a : a A and a
/ B}.

1. Let (X, ) and (Y, ) be topological spaces and let f : X Y be continuous.


(a) Let A be a connected subset of X. Prove that f (A) is a connected subset of Y .
(b) Give an example showing that connected cannot be replaced by closed.
2. Let (X, d) be a metric space.
(a) Let x0 X and let r > 0. Prove that the closed ball B[x0 , r] = {x X : d(x, x0 ) r} is
a closed subset in (X, d).
(b) Prove that X is regular.
3. Let (X, ) and (Y, ) be topological spaces and let f : X Y be continuous.
(a) Assuming that Y is Hausdorff, prove that the graph of f , (f ) = {(x, y) : x X, y = f (x)}
is a closed subset of X Y equipped with the product topology.
(b) Prove that (f ) equipped with the relative topology is homeomorphic to X.
4. Suppose that the topological space (X, ) has a countable base.
(a) Show that if {V : } is a family of open, pairwise disjoint, nonempty subsets of X,
then must be countable.
(b) Let A be an uncountable subset of X. Prove that some point of X must be a limit point
of A. (Hint: if not, consider A as a subset of X.)
5. Let (X, ) be a Hausdorff space. We say that a sequence (xn )
n=1 is convergent to x X iff
for each neighborhood V of x there exists n N such that for each k > n we have xk V . In
this case we write (xn ) x.
(a) Suppose that (xn ) x and (xn ) y. Prove that x = y.
(b) Suppose that (xn ) x. Prove that {xn : n N } {x} is a compact subset of X.
6. The lower limit topology on R, a.k.a. the Sorgenfrey topology, is the topology L having as a
base all half-open intervals [a, b) where a < b.
(a) Is the space (R, L ) first countable? Explain.
(b) Is the space (R, L ) connected? Explain.
(c) Is [0, 1] compact as a subspace of (R, L )? Explain.
7. Let (X, ) be a topological space and (Y, ) be compact topological space. Suppose that F is
a closed subset of X Y and 1 is the usual projection map from X Y to X. Show that if
x0 X \ 1 (F ), then there exists a neighborhood U of x0 such that F (U Y ) = .

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