homotopy and (co)filtered limits

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 ?

• No. For example, take all your $X_i$ to be the $1$-sphere, and all the maps the degree $2$ map. The limit is actually not pathconnected. – Achim Krause Dec 6 '18 at 21:20

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 $$\pi_0(\lim X_i) \to \lim \pi_0(X_i)$$.