Notes:Quotient topology/Table

Table of definitions

Book	Quotient map	Quotient topology	Quotient space	Identification map
An Introduction to Algebraic Topology		Let [ilmath](X,\mathcal{ J })[/ilmath] be a top.. Let [ilmath]X'[/ilmath] denote a partition of [ilmath]X[/ilmath]; and [ilmath]v:X\rightarrow X'[/ilmath] the natural map, [ilmath]v:x\mapsto X_\alpha\in X'[/ilmath] (such that [ilmath]x\in X_\alpha[/ilmath] The quotient topology on [ilmath]X'[/ilmath], [ilmath]\mathcal{K} [/ilmath] is defined as: [ilmath]\forall U\in\mathcal{P}(X')[U\in\mathcal{K}\iff v^{-1}(U)\in\mathcal{J}][/ilmath]		A continuous surjection, [ilmath]f:X\rightarrow Y[/ilmath] is an identification (map) if [ilmath]U\in\mathcal{P}(Y)[/ilmath] is open if and only if [ilmath]f^{-1}(U)[/ilmath] open in [ilmath]X[/ilmath]. If an equivalence relation, [ilmath]\sim[/ilmath] is involved then the "natural map" (canonical projection of an equivalence relation) is an identification
Topology and Geometry	Let [ilmath](X,\mathcal{ J })[/ilmath] be a top., let [ilmath]Y[/ilmath] be a set and [ilmath]f:X\rightarrow Y[/ilmath] a surjective function. The quotient topology on [ilmath]Y[/ilmath] (AKA: topology induced by [ilmath]f[/ilmath]) is defined by: [ilmath]U\in\mathcal{P}(Y)[/ilmath] is open in [ilmath]Y[/ilmath] if and only if [ilmath]f^{-1}(U)[/ilmath] is open in [ilmath]X[/ilmath]		A quotient space is the special case of the quotient topology on [ilmath]X/\sim[/ilmath]
Introduction to Topology (G & G)
Introduction to Topology (Mendelson)
Topology - An Introduction with Applications to Topological Groups
Topology (Munkres)