Difference between revisions of "Notes:Quotient topology/Table"
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! [[Books:Topology and Geometry - Glen E. Bredon|Topology and Geometry]] | ! [[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}} |
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| + | | A '''quotient space''' is the special case of the ''quotient topology'' on {{M|X/\sim}} | ||
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! [[Books:Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene|Introduction to<br/>Topology (G & G)]] | ! [[Books:Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene|Introduction to<br/>Topology (G & G)]] | ||
Latest revision as of 16:09, 13 September 2016
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) |
