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6. Let X be a set define the collection Tfinite C P(X) by OE Tfinite if and only if either O = 0 or X\ O is finite. a) Show that Tfinite is a topology on X. b) What is this topology when X is finite. c) What are the closed subsets of X for the topology Tfinite- d) If X is infinite show that for

6. Let X be a set define the collection Tfinite C P(X) by OE Tfinite if and only if either O = 0 or X\ O is finite. a) Show that Tfinite is a topology on X. b) What is this topology when X is finite. c) What are the closed subsets of X for the topology Tfinite- d) If X is infinite show that for

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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cise 6. Let X be a set define the collection Tfinite CP(X) by OE Tfinite if and only if either O = 0 or X\ O is finite. a) Show that Tfinite is a topology on X. b) What is this topology when X is finite. c) What are the closed subsets of X for the topology Tfinite- d) If X is infinite show that for A, B closed in X, we have X = AUB if and only if either A = X or B = X. e) Assume X infinite. If : X → R is continuous, for the topology Tfinite on X and the usual topology on R, show that is a constant map.
Transcribed Image Text:cise 6. Let X be a set define the collection Tfinite CP(X) by OE Tfinite if and only if either O = 0 or X\ O is finite. a) Show that Tfinite is a topology on X. b) What is this topology when X is finite. c) What are the closed subsets of X for the topology Tfinite- d) If X is infinite show that for A, B closed in X, we have X = AUB if and only if either A = X or B = X. e) Assume X infinite. If : X → R is continuous, for the topology Tfinite on X and the usual topology on R, show that is a constant map.
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