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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.

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It sounds like you are talking about the set of ends of a topological space. There's also a natural way to glue this set to $E$ itself, to create what's called the end compactification of $E$.

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