Difference between revisions of "Homotopy"
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| + | ==Definition== | ||
| + | A ''homotopy'' between two [[topological space|topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}}, is a [[continuous function]]: | ||
| + | * {{M|F:X\times I\rightarrow Y}} (where {{M|I}} denotes the [[unit interval]], {{M|[0,1]\subset\mathbb{R} }}) | ||
| + | A homotopy is ''relative to {{M|A\in\mathcal{P}(X)}}'' if {{M|F(a,t)}} is independent of {{M|t}} for all {{M|a\in A}} | ||
| + | ==Terminology== | ||
| + | The family of functions {{M|\{f_t:X\rightarrow Y\ \vert\ \forall t\in[0,1],\ f_t:x\mapsto F(x,t)\} }} are called the ''stages'' of the homotopy. So we might say: | ||
| + | * Let {{M|f_t}} be a stage of the homotopy {{M|F}} or something similar | ||
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| + | |||
| + | =OLD ATTEMPT AT PAGE= | ||
| + | (Scrapped because I didn't like the layout) | ||
==Definition== | ==Definition== | ||
A ''homotopy'' from the [[topological space|topological spaces]] {{Top.|X|J}} to {{Top.|Y|K}} is a [[continuous function]]{{rATHHRMS}}{{rITTGG}}: | A ''homotopy'' from the [[topological space|topological spaces]] {{Top.|X|J}} to {{Top.|Y|K}} is a [[continuous function]]{{rATHHRMS}}{{rITTGG}}: | ||
Revision as of 20:15, 2 May 2016
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Contents
Definition
A homotopy between two topological spaces, [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath], is a continuous function:
- [ilmath]F:X\times I\rightarrow Y[/ilmath] (where [ilmath]I[/ilmath] denotes the unit interval, [ilmath][0,1]\subset\mathbb{R} [/ilmath])
A homotopy is relative to [ilmath]A\in\mathcal{P}(X)[/ilmath] if [ilmath]F(a,t)[/ilmath] is independent of [ilmath]t[/ilmath] for all [ilmath]a\in A[/ilmath]
Terminology
The family of functions [ilmath]\{f_t:X\rightarrow Y\ \vert\ \forall t\in[0,1],\ f_t:x\mapsto F(x,t)\} [/ilmath] are called the stages of the homotopy. So we might say:
- Let [ilmath]f_t[/ilmath] be a stage of the homotopy [ilmath]F[/ilmath] or something similar
OLD ATTEMPT AT PAGE
(Scrapped because I didn't like the layout)
Definition
A homotopy from the topological spaces [ilmath](X,\mathcal{ J })[/ilmath] to [ilmath](Y,\mathcal{ K })[/ilmath] is a continuous function[1][2]:
- [ilmath]F:X\times I\rightarrow Y[/ilmath] (where [ilmath]I[/ilmath] denotes the unit interval, [ilmath][0,1]\subseteq\mathbb{R} [/ilmath])
For each [ilmath]t\in I[/ilmath] we have a function:
- [ilmath]F_t:X\rightarrow Y[/ilmath] defined by [ilmath]F_t:x\mapsto F(x,t)[/ilmath] - these functions, the [ilmath]F_t[/ilmath] are called the stages[1] of the homotopy.
Applications
References
- ↑ 1.0 1.1 Algebraic Topology - Homotopy and Homology - Robert M. Switzer
- ↑ Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene
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