Spaces

 

is left to the reader.

The respects in which -spaces are `nicer' than others are mostly concerned with `cluster point of a set' (an idea we have avoided!). We show the equivalence, in spaces, of the two forms of its definition used in analysis.

Obviously, (i) (ii); conversely, suppose (i) fails; so there exists a neighbourhood N of p such that is finite. Consider ; it is cofinite and is thus an (open) neighbourhood of p. Hence is a neighbourhood of p which contains no points of A, except possibly p itself. Thus, (ii) fails also.

Hence, (i) (ii).