It is known that if $X$ is a topological vector space (TVS), then all the translations and nontrivial scalar multiplications are homeomorphisms.
I'm curious about the following question about which I have no idea at all: Is the converse also true? More precisely, suppose $X$ is a vector space with a topology $\tau$ such that all the translations and scalar multiplications (with fixed scalar) are homeomorphisms. Can we conclude that $X$ must be a TVS?
Any bijection on a discrete space is automatically a homeomorphism.
A nontrivial vector space over nondiscrete numbers with discrete topology fails to be a TVS.
This gives you a counterexample as follows:
Given a nontrivial vector space over the rational, real or complex numbers. Endow it with the discrete topology. Then translation by any vector and scalar multiplication by nontrivial scalar become homeomorphisms. But scalar multiplication is not even separately nor jointly continuous.