Well, it usually goes the other way. A property $P$ of a topological space $X$ is deserved to be called topological if $P(X)$ holds if and only if $P(Y)$ holds whenever $X$ and $Y$ are homeomorphic. An example of a topological property is "$X$ is connected" while an example of a non-topological property is "$X$ is a subset of $\mathbb{R}^n$. The latter can be upgraded to a topological property by requiring instead "$X$ can be embedded in $\mathbb{R}^n$".
With this convention, given a topological space $X$ there is a smart-ass topological property one can define using $X$: "$P(Z)$ holds iff $Z$ is homeomorphic to $X$". Since homeomorphism is an equivalence relation, this is indeed a topological property and clearly $X$ satisfies $P$. If $Y$ is another space that satisfies the same topological properties as $X$, then $P(Y)$ must hold and so $X$ and $Y$ are homeomorphic.
The domain and range of your functions should have a linear structure to begin with. Then, to talk about limits, your space should have a topology. Topological linear spaces are suite for this.
In normed spaced (Banach spaces for example) there is a notion of total derivative that generalizes the concept of derivative known in Calculus.
Definition (Fréchet) Suppose $X,Y$ are normed spaces, $U\subset X$ open. A function $F:U\longrightarrow Y$ is called differentiable at $x\in U$ if there is $F'(x)\in\mathcal{L}(X,Y)$ such that
$$
F(x+h)=F(x)+F'(x)h + r(h)
$$
where $r(h)=o(h)$; i.e., $\lim_{h\rightarrow0}\frac{|r(h)|}{\|h\|}=0$.
This notion does not generalize to locally convex spaces.
There is another option of differentiability that focusses on directional derivatives which can be generalize to general linear topological spaces, in articular locally convex spaces.
Definition: Suppose $X$ and $Y$ are locally convex linear spaces, $U\subset X$ open. Let function $F:U\rightarrow Y$. The directional derivative of $F$ at $\mathbf{x}\in U$ in the direction $\mathbf{v}$ is defined as
$$
D_vF(\mathbf{x}):=\lim_{t\rightarrow0}\frac{F(\mathbf{x}+t\mathbf{v})-F(\mathbf{x})}{t}
$$
when the limit exists (the limit is with respect to the topology in $Y$, that is for any neighborhood $U$ of $\mathbf{0}\in Y$, there is $\delta>0$ such that if $0<|t|<\delta$, $\frac{F(\mathbf{x+v})-F(\mathbf{x})}{t}\in D_vF(\mathbf{x})+ U$).
Definition: $F$ is said to be Gâteaux--differentiable at $\mathbf{x}\in U$ there is a map $L_x:X\rightarrow Y$ such that $D_vF(\mathbf{x})=L_x\mathbf{v}$ for all $\mathbf{v}\in X$. $L_x$ is called Gâteaux--derivative of $F$ at $\mathbf{x}$.
If $X$ and $Y$ are Banach spaces, and
$F$ is differentiable at $\mathbf{x}\in U$, then $F$ is Gâteaux--differentiable at $\mathbf{x}$ and
$D_vF(\mathbf{x})=F'(\mathbf{x})\mathbf{v}$ for all $\mathbf{v}\in X$. The converse is not necessarily true, unless some continuity conditions on $x\rightarrow L_x$ are satisfied.
Theorem:
Suppose $X$ and $Y$ are Banach spaces, $U\subset X$ open, and let $F:U\subset X\rightarrow Y$ be Gâteaux--differentiable on a neighborhood $V\subset U$ of a point $\mathbf{x}\in U$. If the Gâteaux derivative $y\mapsto L_y\in \mathcal{L}(X,Y)$ \is continuous at $\mathbf{x}$, then $F$ is (Fréchet) differentiable at $\mathbf{x}$ and $F'(\mathbf{x})=L(\mathbf{x})$.
Many books in nonlinear functional analysis (Klaus Deimling's for example) have more details and application of Gâteaux differentiation.
Best Answer
For some classical spaces in topology there has been a long research tradition of finding nice unique characterisations of a space, i.e. a finite list of "real" properties (directly defined in terms of open sets and set theory, like compactness and connectedness (and their local versions) and various ones) such that a space that satisfies that finite list of properties is homeomorphic to the space in question. To be useful, it should be non-circular ones (so being a simple-closed curve is not a "good" property to characterise the circle, because that's already defined as being homeomorphic to a circle, basically).
For zero-dimensional spaces we have the classic ones for the Cantor set (the unique zero-dimensional compact metrisable space without isolated points), the rationals (the only countable metrisable space without isolated points), the irrationals (the only completely metrisable separable zero-dimensional, nowhere locally compact space) and a few more.
For connected metric spaces (often continua) we have classic characterisations of $[0,1]$ (metrisable Peano-continuum with exactly two non-cutpoints), $\mathbb{R}$ (separable metrisable, connected, locally connected and every point a strong cut-point (exactly two components of the space minus that point)), $S^1$( metrisable Peano continuum, with no cutpoints and such that every pair is a cutset) and a few others.
For infinite-dimensional spaces we have the topological classification of completely metrisable separable vector spaces (they're all homeomorphic, so purely topologically there is no difference between $C([0,1])$ and separable Hilbert space, e.g.)
But for general spaces there is no hope of such results, they are just too wild and have lots of connections with complicated set theory. By transfinite recursion one can sometimes construct non-homeomorphic spaces of very similar type (so that they'd be the same for a list of standard properties, at least that I could think of (I did this kind of thing in my PhD thesis) but that nonetheless are non-homeomorphic because they are constructed that way: by a diaginalisation argument one can often enumerate in advance all possible candidate functions, and then during the construction ensure that none of them will be a homeomorphism of the eventually resulting space; it's a common trick topologists use, Sierpinski already did it back in the 1920's). So sometimes spaces are not homeomorpjic because they aren't by construction. And if I made $2^{\aleph_1}$ distinct spaces in some construction, there is no way I could distinguish them even using a finite list from $\aleph_1$ properties, if could invent that many "properties" in the first place.. The only option left would be to give up and allow "homeomorphic to $X$" as a (useless) property for all $X$.
Just my thoughts..