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Why don't we introduce the concept of base for a topology in a minimal way exactly as we did in Linear Algebra?

Edit: A topology can be obtained from a base by considering all possible unions of the basis elements. The way the members of a vector space $V$ are obtained from a spanning set of $V$ is analogous. However in vector space study of smallest of such spanning sets is of primary importance which is not done in Topology.

Linked: Smallest/Minimal bases of a topological space

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Sugata Adhya
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  • Could you be more specific about what you mean by minimal way? – Brian M. Scott Feb 25 '13 at 15:51
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    There is very little advantage in looking for a smallest possible base. In most of the interesting cases they are all infinite anyway. In the development of the theory there is sometimes an advantage in having a countable basis, but often we don't care. Also the process of throwing away basis elements that are not needed in the sense that they are unions of other sets in the basis may never stop. This is in sharp contrast to the process of throwing away those vectors of a generating set that cause linear dependencies. – Jyrki Lahtonen Feb 25 '13 at 15:56
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    @jyrki: In most interesting cases these is no smallest possible base. I recall investigating this a little bit around 1987 or so, and finding that even as general a space as an infinite $T_1$ space has no minimal base under the inclusion ordering. – MJD Feb 25 '13 at 16:17
  • @BrianM.Scott: See Edit. – Sugata Adhya Feb 25 '13 at 17:35
  • The edit doesn’t really help, since it’s not clear what notion of smallest you have in mind. If you mean minimal with respect to inclusion, then as @MJD has already pointed out, there generally isn’t such a base. If you mean minimal with respect to cardinality, the answer is that while the cardinality of a basis for a vector space is of immediate interest in elementary linear algebra, the minimum cardinality of a base for a topology is of importance in elementary general topology only when it’s countable, and usually not even then, as Jyrki pointed out. – Brian M. Scott Feb 25 '13 at 17:44
  • @BrianM.Scott: It's minimal with respect to cardinality. I think even for a vector space basis we are only concerned with minimality with respect to cardinality. – Sugata Adhya Feb 25 '13 at 17:48
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    @MJD, surely you mean there is no smallest base, right? An infinite discrete space (or any discrete space) has a base consisting of singletons, no proper subset of which is a base. – dfeuer Feb 25 '13 at 17:55
  • Honestly I'm not sure what I mean! I haven't thought about it in around 25 years. – MJD Feb 25 '13 at 18:02
  • Also does there exist any relationship between the cardinalities of different bases minimal with respect to inclusion? – Sugata Adhya Feb 25 '13 at 18:14
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    @SugataAdhya a general fact: if $\mathcal{B}$ is a base for $X$, and $\mathcal{B}'$ is another base of $X$, both infinite, then there is a subfamily of $\mathcal{B}'$ that has size at most that of $\mathcal{B}$ and is still a base. So minimal bases with respect to inclusion (if they exist!) we can use as the second base in this statement, and let $\mathcal{B}$ be a base of minimal cardinality. We then see all of such minimal-inclusion bases have the same size = the weight of the space. – Henno Brandsma Feb 25 '13 at 18:29

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