2. Define a cofinite topology J on an infinite set X. Answer each of the following for (X, J): (i) Verify that T is a topology on X; (ii) Show that (X, J) is T₁, but not T2; (iii) Define the concepts of a (a)cover; (b) an open cover; (c) a sub- cover and (d) a finite cover of a set X in a topological space (X, T). Show that the space (X, J), an infinite set with the cofinite topology J, is compact; (iv) Suppose that A is a finite subset of X. Discuss the nature of the subspace (A, JA).
2. Define a cofinite topology J on an infinite set X. Answer each of the following for (X, J): (i) Verify that T is a topology on X; (ii) Show that (X, J) is T₁, but not T2; (iii) Define the concepts of a (a)cover; (b) an open cover; (c) a sub- cover and (d) a finite cover of a set X in a topological space (X, T). Show that the space (X, J), an infinite set with the cofinite topology J, is compact; (iv) Suppose that A is a finite subset of X. Discuss the nature of the subspace (A, JA).
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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Transcribed Image Text:2. Define a cofinite topology J on an infinite set X. Answer each of
the following for (X, J):
(i) Verify that J is a topology on X;
(ii) Show that (X, T) is T₁, but not T2;
(iii) Define the concepts of a (a)cover; (b) an open cover; (c) a sub-
cover and (d) a finite cover of a set X in a topological space (X, J).
Show that the space (X, J), an infinite set with the cofinite topology
J, is compact;
(iv) Suppose that A is a finite subset of X. Discuss the nature of
the subspace (A, JA).
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