Suppose that we have a diagram in the category Top of the following form $$ \cdots X_2 \to X_1 \to X_0=X $$ where the arrows are homeomorphisms. Is it true that that the natural morphism $lim_i X_i \to X$ is a homeomorphism?
It certainly seems like it should be to me, but my intuition sometimes isn't great in these things.
The reason why I am worrying about this is that I wish to show that the infinite barycentric subdivision of a CW complex is homeomorphic to the original complex. Thanks a million in advance for your help!
Yes. The inverse limit $X' = \lim\{X_i, p_i : X_{i+1} \to X_i\}$ comes along with a family of maps $p'_i : X' \to X_i$ such that the universal property of the inverse limit is satisfied (see e.g. https://en.wikipedia.org/wiki/Inverse_limit).
In your case take $X' = X = X_0$ and $p'_i = p_{i-1}^{-1} \circ \ldots \circ p_0^{-1}$ and verify that the universal property is satisfied.
In your question you also ask for the "infinite barycentric subdivision of a CW-complex". I have never heard about the barycentric subdivision of a CW-complex. Usually this only applies to simplicial complexes. But even in that case I have no idea what an "infinite barycentric subdivision" should be. You can form the $n$-th barycentric subdivision for each $n$, but you cannot get something like an $\infty$-th barycentric subdivision. All simplices of such a space would have mesh $0$, i.e. would be points.