Compactness in topology and finite generation in algebra are nice
properties to start with. However, the study of compact spaces
leads naturally to non-compact spaces and infinitely generated
chain complexes; a classical example is the theory of covering
spaces. In handling non-compact spaces we must take into account
the infinity behaviour of such spaces. This necessitates modifying
the usual topological and algebraic cate gories to obtain "proper"
categories in which objects are equipped with a "topologized
infinity" and in which morphisms are compatible with the topology
at infinity. The origins of proper (topological) category theory go
back to 1923, when Kere kjart6 [VT] established the classification
of non-compact surfaces by adding to orien tability and genus a new
invariant, consisting of a set of "ideal points" at infinity.
Later, Freudenthal [ETR] gave a rigorous treatment of the topology
of "ideal points" by introducing the space of "ends" of a
non-compact space. In spite of its early ap pearance, proper
category theory was not recognized as a distinct area of topology
until the late 1960's with the work of Siebenmann [OFB], [IS],
[DES] on non-compact manifolds.
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