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# path

Let $I=[0,1]\subset\mathbb{R}$ and let $X$ be a topological space.

A continuous map $f:I\rightarrow X$ such that $f(0)=x$ and $f(1)=y$ is called a *path* in $X$. The point $x$ is called the initial point of the path and $y$ is called its terminal point. If, in addition, the map is one-to-one, then it is known as an arc.

Sometimes, it is convenient to regard two paths or arcs as equivalent if they differ by a reparameterization. That is to say, we regard $f\colon I\to X$ and $g\colon I\to X$ as equivalent if there exists a homeomorphism $h\colon I\to I$ such that $h(0)=0$ and $h(1)=1$ and $f=g\circ h$.

If the space $X$ has extra structure, one may choose to restrict the classes of paths and reparameterizations. For example, if $X$ has a differentiable structure, one may consider the class of differentiable paths. Likewise, one can speak of piecewise linear paths, rectifiable paths, and analytic paths in suitable contexts.

The space $X$ is said to be pathwise connected if, for every two points $x,y\in X$, there exists a path having $x$ as initial point and $y$ as terminal point. Likewise, the space $X$ is said to be arcwise connected if, for every two distinct points $x,y\in X$, there exists an *arc* having $x$ as initial point and $y$ as terminal point.

A pathwise connected space is always a connected space, but a connected space need not be path connected. An arcwise connected space is always a pathwise connected space, but a pathwise connected space need not be arcwise connected. As it turns out, for Hausdorff spaces these two notions coincide — a Hausdorff space is pathwise connected iff it is arcwise connected.

## Mathematics Subject Classification

54D05*no label found*

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## Comments

## announcement: my trip to Africa

This Saturday my group's going to Africa on a special humanitarian mission involving women's health and women's rights. I can't give any more details than that, some religious people disapprove of what we're doing and we've received death threats.

They tell me I'm gonna have Internet access in my hotel at all times, but even so I'm gonna be out in the villages most of the time, so I probably won't be logging on to this website at all for the next three months. I will try to check my Yahoo! e-mail every chance I get, but I can't make promises on that either. If you spot some serious mistake in one of my entries (something that causes wrong values) please do file a correction but understand that I may take even longer than usual to get around to fixing it.

Wish me bon voyage!

## Re: announcement: my trip to Africa

Have a fun and safe time!

## Re: announcement: my trip to Africa

Bon voyage!

## Re: announcement: my trip to Africa

Bon voyage! I wish that your work gives good results.

Jussi