Neil has already given adequate reply; this answer is partly for Simon, and partly for those who *do* like category theory, and realize that its purpose is to make life simpler, not more complicated!

First, IMHO that's not a very good definition in the wikipedia article. A better definition is given in Spivak's *A Comprehensive Introduction to Differential Geometry, Volume I*, page 30: an **end** of a non-compact topological space $X$ is a function $e$ which assigns to each compact subset $K \subset X$ a nonempty component $e(K)$ of the complement $X - K$, in such a way that $K \subset K'$ implies $e(K') \subset e(K)$. This way of putting it circumvents having to *choose* a covering by interiors of compact sets at the outset, and then requiring a lemma that shows independence of choice.

In categorical language, the set of ends of $X$ is the inverse limit of sets

$$\lim_{K \subset X} \pi_0(X - K)$$

where $K$ ranges over compact subsets.

Anyway, in answer to the question, the point is that any sequence of compact subsets whose interiors cover $X$ is cofinal in the directed set of all compact subsets. (A partially ordered set is *directed* if it is nonempty and if any two elements have an upper bound. A subset is *cofinal* if any element in the partial order is bounded above by an element in the subset.)

The point then is that the limit over a directed set is isomorphic to the limit over a cofinal subset (with partial order inherited from the order of the directed set): in the present case, the sequence $K_j$ is cofinal, and the map given by restriction

$$\lim_{K} \pi_0(X - K) \to \lim_j \pi_0(X - K_j)$$

is a bijection. The inverse function takes a sequence of components $C_j$, and assigns to it the function whose value at $K$ is the unique component of $X - K$ which contains $C_j$, where $K_j$ is any compact subset containing $K$. This doesn't depend on $j$, and it is routine to show this does give the inverse function, according to what Neil has already explained.

But it is really just a special case of a much more general argument about cofinal functors; see Categories for the Working Mathematician, page 217, for a rather more general statement.