# Projective limit and connected components

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 ?

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I've heard the name "ends" (of $E$) for the elements of your $E_\infty$, but I'm not sure how standard that name is. –  Andreas Blass Jan 7 '14 at 19:44
Thanks, it's the name. –  user12806 Jan 7 '14 at 20:05

There is a survey article (by me) that looks at the area of the homotopy theory of ends (and Proper Homotopy) that appeared in the Handbook of Algebraic Topology (1995). If the space concerned is reasonably nice (e.g. an infinite simplicial complex or similar) then the space of ends has a neat interpretation as rays going to $\infty$'. There is a large amount of theory relating to the homotopy invariants of the ends of spaces. This started with Siebenmann's work on finding when an open manifold could be thought of as a closed manifold with the boundary removed.

There are, of course, complications and some of these are discussed in that Survey article and the papers refered to there. That built on an approach through the category of prosimplicial sets due to Edwards and Hastings. A slightly different approach was used by Baues and Quintero in their book Infinite Homotopy Theory'.

Hughes and Ranicki wrote another book `Ends of Complexes' (C.U.Press, 1996), but I have not a copy but there is a pdf file on the web page of Andrew Ranicki. (http://www.maths.ed.ac.uk/~aar/books/ends.pdf).

One of the interesting problems is seeing how the global homotopy type reflects the homotopy type at the ends.

(If you are interested in following some of this up, post another question and I will look out some more precise references. On the other hand, you may be more interested in the general topology of these end spaces, and again there is more to say there as well.)

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I'd also recommend Guilbault's survey arxiv.org/abs/1210.6741 and Ross Geoghegan's book "Topological methods in group theory". –  Michał Kukieła Jan 8 '14 at 13:52
Moreover, it is perhaps worth knowing that there are people investigating ends from the point of view of graph theory, see for example math.uni-hamburg.de/home/diestel/papers/TopGrProject.html . –  Michał Kukieła Jan 8 '14 at 14:04
Thanks, Michael, I did not know of Guibault's survey which loos good (even if he does not seem to know of a lot of the proper homotopy theory that I mentioned!!!) –  Tim Porter Jan 9 '14 at 17:00