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?


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.

  • 1
    $\begingroup$ Are you reading it in English? $\endgroup$
    – user239203
    Dec 27 '19 at 11:49
  • $\begingroup$ 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. $\endgroup$
    – A. Goodier
    Dec 27 '19 at 11:54
  • $\begingroup$ > Are you reading it in English? Yes. $\endgroup$
    – affibern
    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.

So in practice these notions coincide, but there are marginal examples where the notions differ.

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.

  • 2
    $\begingroup$ It is worth highlighting again that for Hausdorff spaces both concepts agree. Then, if one deals with manifolds or more generally with CW-complexes there is no difference at all. $\endgroup$
    – Dog_69
    Dec 27 '19 at 15:46

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