Is it possible to capture the concept of a subspace in Top? By that I mean the following: You are given a morphism $i:Y\to X$ in Top. Is it possible to tell if $Y$ can be regarded as subspace of $X$ with $i$ being the embedding map where you only may use the information the category Top can provide? So you have access to all objects and morphisms but you don't have the functor sending them to Set. So in particular you can't access the elements of a topological space and the open sets.
For sure such an $i$ has to be injective. This can be checked by simply checking if $i$ is monic (which is a category theoretic property). But only knowing that $i$ is injective is not enough because $Y$ could have a much finer topology than the actual subspace. So what is left is checking if $Y$ has the coarsest topology making $i$ continuous. And this has to be done by means of Top.
To get subspace embeddings, instead of monomorphisms, you have to consider extremal / regular monomorphisms.
Recall that a regular monomorphism is a monomorphism that arises as an equalizer. For a subspace inclusion $X\subset Y$ you may consider two arrows $f,g\colon Y \to \{ 0,1 \}$, where $\{ 0, 1 \}$ carries the indiscrete topology, $f$ maps everything to $1$ and $g$ maps to $1$ the elements of $X$. Then the inclusion $i\colon X\hookrightarrow Y$ is the equalizer of $f$ and $g$.
In the other direction, any equalizer of two arrows $f,g\colon Y\to Z$ is some subspace $X\subset Y$.