# Are open sets determined by paths?

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?

• The $p$-adics (or any non-discrete, totally disconnected space) admit no non-constant paths. Feb 4, 2020 at 2:48
• Nice. And the topologist's sine curve will become disconnected in this topology, so maybe we want to say locally path-connected at least. Feb 4, 2020 at 2:51
• Any topology on $[0,1]$ strictly coarser than the usual one gives a counterexample - and e.g. the cofinite topology on $[0,1]$ is path connected. I think the right context for this question, at least at first, is (locally) path-connected Hausdorff spaces. Feb 4, 2020 at 3:10
• The spaces satisfying this condition are often called “delta generated” spaces. Feb 4, 2020 at 3:42
• What about locally path connected spaces? Feb 4, 2020 at 5:06

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.