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• A is open in X. • A = Int A • A contains none of its boundary points. • Every point of A has a neighborhood contained in A.

• A is open in X. • A = Int A • A contains none of its boundary points. • Every point of A has a neighborhood contained in 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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Proposition 8. Let X be a topological space and let ACX be any subset. (a) A point is in Int A if and only if it has a neighborhood contained in A. (b) A point is in Ext A if and only if it has a neighborhood contained in X\A. (c) A point is in ÔA if and only if every neighhorhood of it contains both a point of A and a point of X\A (d) A point is in Ā if and only if every neighborhood of it contains a point of A. (e)Ā = AU DA = Int A U ðA (f) Int A and Ext A are open in X, while Ā and dA are closed in X. (8) The following are equivalent: (Prove:) • A is open in X. • A = Int A • A contains none of its boundary points. • Every point of A has a neighborhood contained in A. (h) The following are equivalent: (Prove:) • A is closed in X. • A = Ā • A contains all of its boundary points. • Every point of X \A has a neighhorhood contained in X\A.
Transcribed Image Text:Proposition 8. Let X be a topological space and let ACX be any subset. (a) A point is in Int A if and only if it has a neighborhood contained in A. (b) A point is in Ext A if and only if it has a neighborhood contained in X\A. (c) A point is in ÔA if and only if every neighhorhood of it contains both a point of A and a point of X\A (d) A point is in Ā if and only if every neighborhood of it contains a point of A. (e)Ā = AU DA = Int A U ðA (f) Int A and Ext A are open in X, while Ā and dA are closed in X. (8) The following are equivalent: (Prove:) • A is open in X. • A = Int A • A contains none of its boundary points. • Every point of A has a neighborhood contained in A. (h) The following are equivalent: (Prove:) • A is closed in X. • A = Ā • A contains all of its boundary points. • Every point of X \A has a neighhorhood contained in X\A.
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