Talk:Topological space

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 Definition:  A mathematical structure (generalizing some aspects of Euclidean space) defined by a family of open sets. [d] [e]
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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:

${\displaystyle X\cap \bigcap K\in {\mathcal {T}}}$
for arbitrary finite family   ${\displaystyle K\subset {\mathcal {T}}}$

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 of 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 ${\displaystyle X}$ -- was false and harmful to the readers of Citizendium. Wlodzimierz Holsztynski 01:20, 18 December 2007 (CST)

Yes, good call. Please see this remark on your talkpage. Hendra I. Nurdin 01:25, 18 December 2007 (CST)

More dramatically, even the uniform spaces do not quite catch the notion of two points being near one another. Indeed, in the case of compact spaces there is no difference between topology and uniformity. Thus my example of homogenuity of manifolds applies also to compact manifolds with their uniform structure. In the non-compact case there is a difference. Consider the partial sums H_n of the harmonic series. The consecutive two sums, H_n and H_(n+1) are arbitrarily close. After applying homeomorphism exp this is no more true for the images exp(H_n) and exp(H_(n+1)) since

lim (exp(H_(n+1) - exp(H_n))   =   1

when n approaches infinity. Thus uniformity makes a difference. (All this is very well known, it's all very basic). Wlodzimierz Holsztynski 05:19, 27 December 2007 (CST)