Difference between revisions of "Notes:Quotient topology/Table"

Latest revision as of 16:09, 13 September 2016

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: $U\in\mathcal{P}(Y)$ is open in $Y$ if and only if $f^{-1}(U)$ is open in $X$		A quotient space is the special case of the quotient topology on $X/\sim$
Introduction to Topology (G & G)
Introduction to Topology (Mendelson)
Topology - An Introduction with Applications to Topological Groups
Topology (Munkres)

@@ Line 18: / Line 18: @@
 |-
 ! [[Books:Topology and Geometry - Glen E. Bredon|Topology and Geometry]]
-| Let {{Top.|X|J}} be a [[topological space|top.]], let {{M|Y}} be a [[set]] and {{M|f:X\rightarrow Y}} a ''[[surjective]]'' [[function]]. The '''quotient [[topology]]''' on {{M|Y}} ({{AKA}}: ''topology induced by {{M|f}}'') is defined by:
+| Let {{Top.|X|J}} be a [[topological space|top.]], let {{M|Y}} be a [[set]] and {{M|f:X\rightarrow Y}} a ''[[surjective]]'' [[function]]. The '''quotient [[topology]]''' on {{M|Y}} ({{AKA}}: ''topology induced by {{M|f}}'') is defined by: {{M|U\in\mathcal{P}(Y)}} is [[open set|open]] in {{M|Y}} {{iff}} {{M|f^{-1}(U)}} is open in {{M|X}}
+|
+| A '''quotient space''' is the special case of the ''quotient topology'' on {{M|X/\sim}}
+|
 |-
 ! [[Books:Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene|Introduction to<br/>Topology (G & G)]]