Let f: XY be a continuous map between topological spaces. We say f is open if for every open subset UCX, 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 AX 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 AnU with U open in X). Exercise: (e) On the other hand, for quotient maps : XX/T corresponding to actions of groups 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 t: A X of subsets A of X that are neither empty nor X itself?
Let f: XY be a continuous map between topological spaces. We say f is open if for every open subset UCX, 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 AX 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 AnU with U open in X). Exercise: (e) On the other hand, for quotient maps : XX/T corresponding to actions of groups 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 t: A X of subsets A of X that are neither empty nor X itself?
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,
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 : 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 π : 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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