Notes:Quotient topology/Table

Table of definitions

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