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?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 21E: Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by...

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