Are open sets determined by paths?

Question

Let $X$ be a topological space, let $U \subset X$, and suppose that for every path $\gamma\colon [0,1] \to X$ the preimage $\gamma^{-1}(U)$ is open. Is it true that $U$ is open? Presumably not in general, but are there reasonable requirements we can put on $X$ to make it true?

To put it another way, the subsets $U$ as above give a topology on $X$, which is the same as or finer than the original topology. When is it the same?

To put it a third way, consider the evaluation map $I \times C(I,X) \to X$, where $I = [0,1]$ and $C(I,X)$ is the set of continuous maps, with (unusually) the discrete topology. When is this is a quotient map?

Answer 1

A space $X$ is called $\Delta$-generated if $U$ is open in $X$ if and only if $\alpha^{-1}(U)$ is open in $[0,1]$ for every path $\alpha:[0,1]\to X$.

It's easy to see that a space $X$ is $\Delta$-generated if and only if $X$ is a quotient space of a disjoint union of copies of $[0,1]$. It follows that every $\Delta$-generated space is a locally path-connected $k$-space. It's also not too hard to show that every first countable locally path-connected space is $\Delta$-generated. Hence,

$$\text{locally path connected & first countable}\Rightarrow \Delta\text{-generated}\Rightarrow\text{locally path connected & }k\text{-space}$$

The category of $\Delta$-generated spaces is very well-behaved, namely, it is a Cartesian closed coreflective subcategory of $\mathbf{Top}$.

The following ``directed hedgehog space" is a locally path-connected space that is not $\Delta$-generated: Let $(A_{\lambda},a_{\lambda})$ be a copy of the based unit interval $([0,1],0)$ for all ordinals $\lambda<\omega_1$. Consider the one-point union $X=\bigvee_{\lambda<\omega_1}A_{\lambda}$ with based point $x_0$ and with the directed wedge topology. This means that $U$ is open in $X$ if $U\cap A_{\lambda}$ is open in $A_{\lambda}$ for all $\lambda$ and if when $x_0\in U$, there exists $\kappa<\omega_1$ such that $A_{\lambda}\subseteq U$ for all $\lambda\geq \kappa$. Clearly, $X$ is locally path-connected and as the quotient of a compact Hausdorff space, $X$ is a $k$-space. Let $b_{\lambda}$ be the image of $1$ in $A_{\lambda}$, i.e. the end of the $\lambda$-th arc, and set $C=\{b_{\lambda}\mid \lambda<\omega_1\}$. Since $[0,1]$ is sequentially compact and no sequence of countable ordinals converges to $\omega_1$, one can show that every path in $X$ intersects $C$ in at most finitely many points. Even though $C$ is not closed in $X$, $\alpha^{-1}(C)$ is closed in $X$ for every path $\alpha:[0,1]\to X$.

By similar reasoning, the quotient space $\displaystyle X'\cong \frac{(\omega_1+1)\times [0,1]}{\{\omega_1\}\times [0,1]\cup (\omega_1+1)\times\{0\}}$ will be a non-$\Delta$-generated $k$-space but it is not locally path-connected. I imagine that you can build a non-$\Delta$-generated space that is both locally path-connected and a $k$-space, but I don't know one off the top of my head. The $k$-space coreflection of $X$ and locally-path connected coreflection of $X'$ don't seem to work.