Suppose we have a (co)filtered digaram $\dots \rightarrow X\_{2}\rightarrow X\_{1}$ of topological space. Is is true that the natural map $\pi\_{0}[\lim X\_{i}]\rightarrow \lim \pi\_{0}(X\_{i})$ is an isomorphism ?



homotopy and (co)filtered limits
This is not true, for two distinct reasons.


1. The first is that the inverse system of spaces may not behave well homotopy-theoretically. If $X\_n = [n, \infty) \subset \Bbb R$, then the limit of $\dots \to X\_2 \to X\_1 \to X\_0$ is empty. However, on path components it is the constant system $\dots \to \* \to \* \to \*$, with limit $\*$. Roughly, you might have a path component that is represented by any space $X\_i$ that is not represented by any compatible system of points. This might make $\pi\_0 \lim X\_i \to \lim \pi\_0 X\_i$ not surjective.
2. The second is the opposite: the map $\pi\_0 \lim X\_i \to \lim \pi\_0 X\_i$ may not be injective. In this case, you may have two points $x$ and $y$ in the limit such that the images in any individual $x\_i$ are connected by a path, but where no path can be compatibly lifted all the way up the tower. For example, if $f:S^1 \to S^1$ is a degree-2 covering map, then the limit of the tower $$\dots \xrightarrow{f} S^1 \xrightarrow{f} S^1 \xrightarrow{f} S^1$$ is called the 2-adic solenoid and it has uncountably many path components.


The first problem goes away if the maps $X\_i \to X\_{i-1}$ are fibrations, and in this case we often call the limit a *homotopy limit*. The second problem does not go away in this case, but Milnor proved that $\pi\_0 \lim X\_i$ is built out of two terms: $\lim(\pi\_0 X\_i)$ and a second term called $\lim^1(\pi\_1 X\_i)$. In particular, if the spaces $X\_i$ are simply-connected then there are no contributions from the second term, and so there will be an isomorphism $\pi\_0(\lim X\_i) \to \lim \pi\_0(X\_i)$.


