Talk:Topological space

Bourbaki convention and topological axioms

If property of the openness of the empty set and of the whole space is included in the set of axioms then the axiom about the intersection of open sets should elegantly state the case of two sets only. On the other hand Bourbaki has ommitted the axiom of the openness of the empty set and of the total space. Instead, Bourbaki introduced the stronger axiom about the openness of the arbitrary finite intersection of open sets. Bourbaki also assumes the openness of the arbitrary union of the open sets. "Arbitrary" in both cases includes the empty case, i.e. the respective operation on the empty set od open subsets of the space. The union of the empty family is the empty set--that's the first Bourbaki convention, and a very reasonable one. The other Bourbaki convention is a bit less clean: the intersection of the empty family of subsets (as opposed to sets) of X is the whole X. Thus Bourbaki, without making any explicit apology, considers not the customary operation of the intersection of sets but an intersection operation which depends on X--it differs from the customary operation only when the family is empty. The customary intersection of the empty family is either the class (not a set) of all sets, or it is not defined, depending on the foundations of mathematics which are applied. My own way out of this dilemma was to go along the Bourbaki 2-axiom approach, except for a modification of the intersection axiom:

$X\cap \bigcap K\in {\mathcal {T}}$

for arbitrary finite family   $K\subset {\mathcal {T}}$

Now we don't have to worry about the foundations. Wlodzimierz Holsztynski 21:23, 17 December 2007 (CST)

Nearness

As a rule, topology does not tell us which points are near one to another. For instance, in the most important case of connected manifolds of dimension > 1, for every two pairs of different points there exists a homeomorphism which maps one pair onto the other. Thus there is not a trace of the notion of "close one to another". For this we have the notion of uniform spaces, of the nearness structures, and also of metrics and of systems semi-metrics. The claim (in the very introduction!) about topology as a general mean of providing the notion of -- the "closeness" of two elements of $X$ -- was false and harmful to the readers of Citizendium. Wlodzimierz Holsztynski 01:20, 18 December 2007 (CST)