Theorem 4.11: Let f: X → Y be a function on the indicated topological spaces. The following statements are equivalent. (1) ƒis continuous. (2) For each closed subset C of Y, ƒ-'(C) is closed in X. (3) For each subset A of X, f(Ã) C JA). (4) There is a basis B for the topology of Y such that f~'(B) is open in X for each basic open set B in B. (5) There is a subbasis 8 for the topology of Y such that f-'(S) is open in X for each subbasic open set S in S.
Theorem 4.11: Let f: X → Y be a function on the indicated topological spaces. The following statements are equivalent. (1) ƒis continuous. (2) For each closed subset C of Y, ƒ-'(C) is closed in X. (3) For each subset A of X, f(Ã) C JA). (4) There is a basis B for the topology of Y such that f~'(B) is open in X for each basic open set B in B. (5) There is a subbasis 8 for the topology of Y such that f-'(S) is open in X for each subbasic open set S in S.
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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Question: prove (2) iff (5)
Hint: use definition of basis from (principles of Topology book)
Transcribed Image Text:Theorem 4.11: Let f: X → Y be a function on the indicated topological spaces.
The following statements are equivalent.
(1) fis continuous.
(2) For each closed subset C of Y, ƒ¯'(C) is closed in X.
(3) For each subset A of X, f{Ã) CJA).
(4) There is a basis B for the topology of Y such that f~'(B) is open in X for
each basic open set B in B.
(5) There is a subbasis S for the topology of Y such that f -'(S) is open in X
for each subbasic open set S in S.
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