Topological and Metric Spaces
DOI link for Topological and Metric Spaces
Topological and Metric Spaces book
When introducing the concept of topology, one faces the common problem of the choice of a particular path
of reasoning, or equivalently, the particular deﬁnition of topology. Mathematics is full of such logical or
rather didactic problems. When two statements describing properties of the same object are equivalent to
each other, then one can be selected as a deﬁnition, whereas the other can be deduced as a consequence.
For instance, we may call upon the equivalence of the Axiom of Choice and the Kuratowski-Zorn Lemma
discussed in Chapter 1. The two statements are equivalent to each other and, indeed, it is a matter of a purely
arbitrary choice that the Axiom of Choice bears the name of an axiom while the Kuratowski-Zorn Lemma
serves as a theorem. One of course may argue that it is easier to accept intuitively the Axiom rather than the
Lemma, but such reasoning has very little to do with (formal) logic, of course.