# The composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths

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## Contents

## Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces, let [ilmath]\varphi:X\rightarrow Y[/ilmath] be a continuous map and let [ilmath]f_1,f_2:I\rightarrow X[/ilmath] be paths in [ilmath]X[/ilmath] such that:

- [ilmath]f_1\simeq f_2\ (\text{rel }\{0,1\})[/ilmath] (that is to say [ilmath]f_1[/ilmath] and [ilmath]f_2[/ilmath] are end-point-preseriving homotopic)

then^{[1]}:

- [ilmath](\varphi\circ f_1)\simeq(\varphi\circ f_2)\ (\text{rel }\{0,1\})[/ilmath]

That is to say:

- The relation of paths being end-point-preseriving-homotopic is preserved under composition with continuous maps.

## Purpose

This is a precursor theorem to:

## Proof

We need to show that [ilmath]H':\ (\varphi\circ f_1)\simeq(\varphi\circ f_2)\ (\text{rel }\{0,1\})[/ilmath], we will do this by constructing a homotopy, [ilmath]H':I\times I\rightarrow Y[/ilmath], between them. We know that:

- [ilmath]H:f_1\simeq f_2\ (\text{rel }\{0,1\})[/ilmath], where [ilmath]H:I\times I\rightarrow X[/ilmath] is the homotopy between the paths [ilmath]f_1[/ilmath] and [ilmath]f_2[/ilmath].

Define:

- [ilmath]H':I\times I\rightarrow Y[/ilmath] by [ilmath]H':(u,t)\mapsto \varphi(H(u,t))[/ilmath], or more explicitly: [ilmath]H':\eq\varphi\circ H[/ilmath]

Grade: C

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It remains to be shown that [ilmath]H'[/ilmath] is a homotopy between [ilmath](\varphi\circ f_1)[/ilmath] and [ilmath](\varphi\circ f_2)[/ilmath] we must show that the initial and final stage of the homotopy is equal to them, this is really not difficult!

## See also

## References

Categories:

- Stub pages
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- Theorems
- Theorems, lemmas and corollaries
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- Topology Theorems, lemmas and corollaries
- Topology
- Algebraic Topology Theorems
- Algebraic Topology Theorems, lemmas and corollaries
- Algebraic Topology
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- Homotopy Theory Theorems, lemmas and corollaries
- Homotopy Theory