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!