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Consider the topology T on X= (a, b, c, d, e), (X, Ø, (a), (e,d}, {a, e, d), (b, c, d, e}} Show that by an example (a) AnaA = 0 and à = ÅU OA; (b) A = 0 if and only if A is both open and closed; (c) A is open if and only if 8A =A\A.

Consider the topology T on X= (a, b, c, d, e), (X, Ø, (a), (e,d}, {a, e, d), (b, c, d, e}} Show that by an example (a) AnaA = 0 and à = ÅU OA; (b) A = 0 if and only if A is both open and closed; (c) A is open if and only if 8A =A\A.

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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Consider the topology T H on X (a, b, c, d, e), (X, Ø, (a), (c,d), (a, e, d), (b, c, d, e}} Show that by an example (a) AnaA = 0 and à = ÅU OA; (b) A = 0 if and only if A is both open and closed; (c) A is open if and only if 8A =A\A.
Transcribed Image Text:Consider the topology T H on X (a, b, c, d, e), (X, Ø, (a), (c,d), (a, e, d), (b, c, d, e}} Show that by an example (a) AnaA = 0 and à = ÅU OA; (b) A = 0 if and only if A is both open and closed; (c) A is open if and only if 8A =A\A.
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Step 1

It is given that,  X = {a, b, c, d, e} and  T = X, ∅, a, c, d, a, c, d, b, c, d, e be a topology on X.

We have to show the following by an example ;

(a) A∘∩∂A = ∅  and  A¯ = A∘∪∂A

(b) ∂A = ∅ if and only if  A is both open and closed.

(c) A is open if and only if ∂A = A¯\A.

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