Difference between revisions of "Homotopy"
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==Definition== | ==Definition== | ||
Given two [[topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}} then a ''homotopy of maps (from {{M|X}} to {{M|Y}})'' is a ''[[continuous]]'' [[function]]: {{M|F:X\times I\rightarrow Y}} (where {{M|I}} denotes the [[unit interval]], {{M|1=I:=[0,1]\subset\mathbb{R} }}). Note: | Given two [[topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}} then a ''homotopy of maps (from {{M|X}} to {{M|Y}})'' is a ''[[continuous]]'' [[function]]: {{M|F:X\times I\rightarrow Y}} (where {{M|I}} denotes the [[unit interval]], {{M|1=I:=[0,1]\subset\mathbb{R} }}). Note: | ||
| − | * The ''stages of the homotopy, {{M|F}},'' are a family of functions, {{M|\{ f_t:X\rightarrow Y\ \vert\ t\in[0,1]\} }} such that {{M|f_t:x\rightarrow F(x,t)}}. [[The stages of a homotopy are continuous]]. | + | * The '''''stages of the homotopy, {{M|F}},''''' are a family of functions, {{M|\{ f_t:X\rightarrow Y\ \vert\ t\in[0,1]\} }} such that {{M|f_t:x\rightarrow F(x,t)}}. [[The stages of a homotopy are continuous]]. |
** {{M|f_0}} and {{M|f_1}} are examples of stages, and are often called the ''initial stage of the homotopy'' and ''final stage of the homotopy'' respectively. | ** {{M|f_0}} and {{M|f_1}} are examples of stages, and are often called the ''initial stage of the homotopy'' and ''final stage of the homotopy'' respectively. | ||
Two ([[continuous]]) functions, {{M|g,h:X\rightarrow Y}} are said to be ''homotopic'' if there exists a homotopy such that {{M|1=f_0=g}} and {{M|1=f_1=h}} | Two ([[continuous]]) functions, {{M|g,h:X\rightarrow Y}} are said to be ''homotopic'' if there exists a homotopy such that {{M|1=f_0=g}} and {{M|1=f_1=h}} | ||
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Contents
[hide]Definition
Given two topological spaces, (X,J) and (Y,K) then a homotopy of maps (from X to Y) is a continuous function: F:X×I→Y (where I denotes the unit interval, I:=[0,1]⊂R). Note:
- The stages of the homotopy, F, are a family of functions, {ft:X→Y | t∈[0,1]} such that ft:x→F(x,t). The stages of a homotopy are continuous.
- f0 and f1 are examples of stages, and are often called the initial stage of the homotopy and final stage of the homotopy respectively.
Two (continuous) functions, g,h:X→Y are said to be homotopic if there exists a homotopy such that f0=g and f1=h
Notes
- Jump up ↑ Do not shorten this to "homotopy equivalence" as homotopy equivalence of spaces is something very different
References
Template:Algebraic topology navbox
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