Definition:Path (Topology)

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Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $I \subset \R$ be the closed real interval $\closedint a b$.


A path in $T$ is a continuous mapping $\gamma: I \to S$.


The mapping $\gamma$ can be referred as:

a path (in $T$) joining $\map \gamma a$ and $\map \gamma b$

or:

a path (in $T$) from $\map \gamma a$ to $\map \gamma b$.


It is common to refer to a point $z = \map \gamma t$ as a point on the path $\gamma$, even though $z$ is in fact on the image of $\gamma$.


Initial Point

The initial point of $\gamma$ is $\map \gamma a$.

That is, the path starts (or begins) at $\map \gamma a$.


Final Point

The final point of $\gamma$ is $\map \gamma b$.

That is, the path ends (or finishes) at $\map \gamma b$.


Endpoint

The initial point and final point of $\gamma$ can be referred to as the endpoints of $\gamma$


Composable Paths

Let $f, g: \closedint 0 1 \to T$ be paths.


$f$ and $g$ are said to be composable paths if:

$\map f 1 = \map g 0$.


Also defined as

The definition for path as given here is usually used in this form in complex analysis, where the details of the mapping itself tend to be important.

However, in topology it is often the case that all that is needed is merely to demonstrate the existence of such a mapping.

Thus it is usual in topology to specify $I \subset \R$ to be the closed unit interval $\closedint 0 1$, and to focus attention on the image of $I$.

From Closed Real Intervals are Homeomorphic the two definitions are seen to be equivalent.


Also known as

Some sources refer to a path as an arc.

However, the word arc, when used in the context of topology, is specifically defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ to mean an injective path.

It is preferred that the distinction remains.


Also see

  • Results about paths in the context of topology can be found here.


Sources

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