Btukfyl

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

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