# Arcwise connected vs. path connected

I've been reading An Introduction to Algebraic Topology by Andrew Wallace.

His definition of an arcwise connected space is: a space in which there is a path between every two points.

In the Wikipedia page for connected space, however, this is taken as the definition of a path-connected space not an arcwise connected space.

I am confused by this mix-up. Which one is more standard?

## Update

I'm quoting Wallace's definition of path and arcwise connectedness.

Definition. Let $$E$$ be a given topological space, and let $$I$$ denote the unit interval $$0 \le t \le 1$$, regarded as a subspace of the space of real numbers in the usual topology. Then a path in $$E$$ joining two points $$p$$ and $$q$$ of $$E$$ is defined to be a continuous mapping $$f$$ of $$I$$ into $$E$$ such that $$f(0) = p$$ and $$f(1) = q$$. The path will be said to lie in a subset $$A$$ of $$E$$ if $$f(I) \subset A$$.

Definition. A topological space $$E$$ is said to be arcwise connected if, for every pair of points $$p$$ and $$q$$ of $$E$$ there is a path in $$E$$ joining $$p$$ and $$q$$. If $$A$$ is a set in a topological space $$E$$, then $$A$$ is arcwise connected if every pair of points of $$A$$ can be joined by a path in $$A$$.

It seems that the situation is as Henno Brandsma pointed out: since most spaces are Hausdorff, path-connectedness is identified with arcwise connectedness and thus entirely omitted from the book. However, this strikes me as odd as I am actually interested in non-Hausdorff spaces such as those arising from computer science.

• Are you reading it in English?
– user239203
Dec 27 '19 at 11:49
• It might also be useful to note that some authors use arcwise connected to mean something stronger - there is a path between every pair of points that is a homeomorphism from $[0,1]$ to its image. Dec 27 '19 at 11:54
• > Are you reading it in English? Yes. Dec 27 '19 at 20:05

Do a good reading of that Wikipedia page: path-connected (usually) means that for every $$a\neq b \in X$$ there is a path from $$a$$ to $$b$$ (a continuous function from $$[0,1]$$ into the space $$X$$ with $$p(0)=a$$ and $$p(1)=b$$ is called a path from $$a$$ to $$b$$), while for an arcwise connected space $$X$$ there is the stronger requirement that there is such a $$p$$ but that also is an embedding (so that $$p:[0,1] \to p[[0,1]]$$ is a homeomorphism). So an arcwise connected space is always path-connected but the reverse sometimes does not hold (there are finite counterexamples, e.g.; obviously an arcwise connected space must be uncountable, while a finite space can be path-connected: a continuous map is a lot weaker than an embedding)
A theorem (not trivial though, it depends on the study of Peano continua) proves that if $$X$$ is Hausdorff (as is very common, e.g. all metric spaces, ordered spaces, many topological vector spaces etc.) then $$X$$ being path-connected implies $$X$$ is arcwise connected too. This can be quite useful to know.
Maybe for your first author path actually means a homeomorphic copy of $$[0,1]$$ and he really defines what most texts would call arcwise connectedness. Check his definition of path.