Consider the two vector spaces $C[0,1]$ and $C[0,1)$, the spaces of continuous functions on $[0,1]$ and $[0,1)$ respectively. Note I do NOT give either of these a topology, I am purely interested in their properties as vector spaces. My question is whether they are isomorphic? The motivation comes from considering $C[0,\infty)$ which is isomorphic to $C[0,1)$. It is very easy to show that $C[a,b]$and $C[c,d]$ are isomorphic for real constants $b>a$, $d>c$ but this got me thinking about this more general case.
Possible further questions would be:
What about $C(0,1)$ (open interval), is this isomorphic to either of the above?
If I consider the space of continuous functions on $[0,\infty)$ that converge to a finite limit at $\infty$ then this is isomorphic to $C[0,1]$ (either compactify the half line or use the usual arctan function to map to $C[0,1)$ and take limits at $1$). But what about the space of bounded continuous functions on the half line?
These cases seem to be easier to look at if we do include topologies but it's interesting that as far as I can tell, it becomes harder without.
The answer is negative without the axiom of choice, and in particular there is no explicit way to define an isomorphism between the various spaces.
It follows from assumptions such as "Every set of reals has the Baire property" (which is consistent without the axiom of choice) that a vector space can have at most one Fréchet topology. This is a consequence of automatic continuity, which gives us that every linear operator between Fréchet spaces is continuous.
If $C(0,1)$ or $C[0,1)$ were isomorphic as vector spaces to $C[0,1]$, this vector space would have several distinct Fréchet topologies, which is a contradiction.