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A Course in Mathematical Analysis by D. J. H. Garling

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13

Topological spaces

13.1 Topological spaces

The results of the previous section show that many important results concerning metric spaces depend only on the topology. We now generalize this, by introducing the notion of a topological space. This is traditionally defined in terms of open sets. A topological space is a set X, together with a collection τ of subsets of X which satisfy:

the empty set and X are in τ ;

if then ;

if O1 and O2 are in τ then O1O2 τ.

Then τ is the topology on X, and the sets in τ are called open sets. The conditions ...

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