Let $E$ be a topological space. Let $\mathcal{K}$ be the set of the compact subsets of $E$.
$(E-K)_{K \in \mathcal{K}}$ is a projective system, because if $K,K'$ are two compacts, there are two inclusion maps $E-(K \cup K') \rightarrow E-K$ and $E-(K \cup K') \rightarrow E-K'$.
Let $F$ be the functor that associates to $E-K$ the set of the connected components of $E-K$.
Let $X$ be a topological space, we define $x \equiv y$ if $x, y \in X$ are in the same connected component in $X$, and we define $\overline{X}$ the quotient of $X$ by $\equiv$.
So $F(E-K)=\overline{E-K}$.
If $Y \subset X$, there is a map $\overline{Y} \rightarrow \overline{X}$.
So $(\overline{E-K})_{K \in \mathcal{K}}$ is a projective system too.
Let $E_{\infty}$ be the projective limit of $(F(E-K))_{K \in \mathcal{K}}$. It's a topological space because $F(E-K)=\overline{E-K}$ has the discrete topology.
What is the name of $E_{\infty}$ ? Do you have references ?
Thanks in advance.