Inverse limit of compact hausdorff spaces

Question

Suppose that $(X,\varphi_i)$ is an inverse limit of an inverse system of compact hausdorff spaces $\{X_i,\varphi_{ij}\}$. Let $Y$ be a subspace of $X$, I would like to proof that $\overline{Y}$ (closure in X) is the inverse limit of $\{\varphi_i(Y),\varphi_{ij}\}$. Since $\varphi_i:Y\rightarrow \varphi_i(Y)$ is onto, the image of the inclusion $Y\rightarrow \text{lim} \ \varphi_i(Y)$ is dense. The problem is that I don't know how to show that $\text{lim} \ \varphi_i(Y)$ is closed in $X$.

Answer 1

This isn’t true. Consider an extremely simple directed system with only one object, with the unique $\phi$ simply being the identity map. Then applying your claim to this system, we get that every subset of every compact Hausdorff space is closed, which is clearly false.