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(e) On the other hand, for quotient maps : X→ X/T corresponding to actions of groups I' on a space X, prove that is automatically open. O R L:11

(e) On the other hand, for quotient maps : X→ X/T corresponding to actions of groups I' on a space X, prove that is automatically open. O R L:11

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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Please show the following questions step by step in detail, please focus on the mathematical writing styles. Its from a topology course,
 
Let f: X Y be a continuous map between topological spaces. • We say f is open if for every open subset UC X, the subset f(U) CY is open. we say f is closed if for every closed subset ACX, the subset f(A)CY is closed. We say that it is a local homeomorphism if for each x € X there exists an open neighborhood U of x in X, and V of (x) in Y such that the restriction of to U becomes a homeomorphism |u: U →V. Given a subset AC X, the inclusion of A into X is the map 1: A → X given by (a) = a for a € A (remember that a function is not just a formula, it consists of three pieces: a domain (which is a set), a codomain (also a set) and a rule (usually a formula) that associates to an element in the domain an element in the codomain). If X is a space then, by default, we endow A with the topology induced from X (opens in A being those of type An U with U open in X). Exercise: (e) On the other hand, for quotient maps 7 : X → X/T corresponding to actions of groups I on a space X, prove that is automatically open. (f) For which spaces X one can still make a statement as in (b), about all inclusions : A → X of subsets A of X that are neither empty nor X itself?
Transcribed Image Text:Let f: X Y be a continuous map between topological spaces. • We say f is open if for every open subset UC X, the subset f(U) CY is open. we say f is closed if for every closed subset ACX, the subset f(A)CY is closed. We say that it is a local homeomorphism if for each x € X there exists an open neighborhood U of x in X, and V of (x) in Y such that the restriction of to U becomes a homeomorphism |u: U →V. Given a subset AC X, the inclusion of A into X is the map 1: A → X given by (a) = a for a € A (remember that a function is not just a formula, it consists of three pieces: a domain (which is a set), a codomain (also a set) and a rule (usually a formula) that associates to an element in the domain an element in the codomain). If X is a space then, by default, we endow A with the topology induced from X (opens in A being those of type An U with U open in X). Exercise: (e) On the other hand, for quotient maps 7 : X → X/T corresponding to actions of groups I on a space X, prove that is automatically open. (f) For which spaces X one can still make a statement as in (b), about all inclusions : A → X of subsets A of X that are neither empty nor X itself?
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