Miscellany

Abstract

At this point, it would be a good idea to briefly reread Chapter 2.6, as we now wish to prove higher-dimensional analogs of those theorems. In all of the theorems of that section the key step is the contradiction obtained when a continuous function tries to take a connected space to a non-connected space. This contradiction could be restated in terms of homology groups, since the group H₀(K) detects connectivity. Only with the introduction of functions on topological spaces and their action on the homology groups can we get results comparing two different spaces.

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“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?”

Albert Einstein