On the openness of the map $X^I \to X \times X.$

Question

Let $X$ be a locale or a topological space. $I$ denote the unit interval of the real numbers, and $X^I$ the space of functions from $I$ to $X$ (The locale exponential if $X$ is a locale or the set of function endowed with the open-compact topology if $X$ is a topological space.)

In both case, we consider the map $X^I \rightarrow X \times X$, which is simply the evaluation at the two endpoints.

In "Connected locally connected toposes are path connected" (available here : http://www.ams.org/journals/tran/1986-295-02/S0002-9947-1986-0833712-3/) Moerdijk and Wraith showed that if $X$ is a locale (or actually even a topos) which is connected and locally connected then the map $X^I \rightarrow X \times X$ is an open surjection.

At the very end of the paper, they mentioned without proof that in the case of topological space, the map $X^I \rightarrow X \times X$ is open if and only if $X$ is "semi-locally path connected".

But if I assume that $X^I \rightarrow X \times X$ is an open map, and $U$ is any open set of $X$, then $U^I$ is an open of $X^I$. hence by restriction, the map $U^I \rightarrow U \times U$ is also an open map. In particular its image which is the relation "$x$ is path connected with $y$ in $U$ " is open in $U \times U$.

So if $x$ is any point of $U$ the path connected component of $x$ in $U$ is the set of $y$ path connected to $x$ in $U$, and hence is open by the previous observation. Finally, any open of $X$ has open path connected component, ie $X$ is locally path connected.

I don't exactly know what is meant by "semi-locally path connected" but I though it was weaker than "locally path connected" so there is something weird here !

My questions are :

• Is there a mistake in what I just said, a mistake in the paper, or am I misunderstanding the notion "semi-locally path connected " ?

• For a general locale, doesn't the openness of the map $X^I \rightarrow X \times X$ actually equivalent to the fact that $X$ is locally connected ?

Thank you !

Answer 1

Here are what I believe are the standard definitions:

A topological space $X$ is locally path-connected if for every $x\in X$ and every open neighbourhood $U$ of $x$, there exists a smaller open neighbourhood $V\subseteq U$ of $x$ such that $V$ is path-connected. Equivalently, path components of open subsets are open.

A topological space $X$ is semi-locally path-connected if for every $x\in X$ and every open neighbourhood $U$ of $x$, there exists a smaller open neighbourhood $V\subseteq U$ of $x$ such that every $y\in V$ is connected to $x$ by a path whose image lies in $U$ (but not necessarily in $V$).

So semi-locally path-connected is formally weaker. But in fact, the definitions turn out to be equivalent for all spaces (this is an exercise on this sheet).