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Exercise 1. Let (X, T) be a topological space. Suppose A is a subset of X, endowed with the subspace topology TA. a) Show that A is open (resp. closed) for T if and only if every B C A which is open (resp. closed) for TA is also open (resp. closed) for T. b) If A is open for T, show that the interior for TA of a subset BCA is the same as its interior for T.

Exercise 1. Let (X, T) be a topological space. Suppose A is a subset of X, endowed with the subspace topology TA. a) Show that A is open (resp. closed) for T if and only if every B C A which is open (resp. closed) for TA is also open (resp. closed) for T. b) If A is open for T, show that the interior for TA of a subset BCA is the same as its interior for T.

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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#1 Need parts a and b
Exercise 1. Let (X,T) be a topological space. Suppose A is a subset of X, endowed with the subspace topology TA- a) Show that A is open (resp. closed) for T if and only if every B C A which is open (resp. closed) for TA is also open (resp. closed) for T. b) If A is open for T, show that the interior for TA of a subset BCA is the same as its interior for T.
Transcribed Image Text:Exercise 1. Let (X,T) be a topological space. Suppose A is a subset of X, endowed with the subspace topology TA- a) Show that A is open (resp. closed) for T if and only if every B C A which is open (resp. closed) for TA is also open (resp. closed) for T. b) If A is open for T, show that the interior for TA of a subset BCA is the same as its interior for T.
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