Does Every Topology Have a Minimal Subset Basis?

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The discussion centers on whether every topology has a minimal subset basis and if Zorn's Lemma is necessary for proving this. A proposed proof using Zorn's Lemma is outlined, but doubts arise regarding its validity. The conversation highlights that while some spaces, like metric spaces, may have minimal subbases, not all topologies necessarily do. A specific counterexample is suggested, indicating that the assertion may not hold universally. The conclusion suggests that the existence of minimal bases is not guaranteed for all topologies.
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I can't seem to find this result in any of my textbooks. Given any basis B for a topology T on X, is there a minimal subset M of B that also is a basis for T (in the sense that any proper subset of M is not a basis for T)? If so, is Zorn's Lemma needed to prove this?
Is the same true of subbases?

Attempt at proof using Zorn's Lemma:
Let B be a basis for a topology T on X. Let A be the collection of all bases for T that is a subcollection of B. A is not empty because B is in A. Partially order A by set containment (i.e. D < E iff D contains E). Let C = {C_i} be a totally ordered subcollection of A. Let K = n(C_i) (intersection). We must show that K is a basis for T. Let U be a T-open set, and let x be in U. Since each C_i is a basis for T, then for each i, there exists C in C_i such that x is in C is a subset of U. Wait, C needs not be the same, and C needs not be in K.

Is the assertion false? What's a counterexample?
 
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How about this weaker assertion: Every topology T contains a minimal basis B for T (in the sense that any proper subset of B is not a basis for T).

This must be true, right? And the same for subbases? But Zorn's Lemma still doens't work.
 
My question even stumped topologist Henno Bradsma. He said:

"I found a result that a metric space has a minimal subbase (proved by van Emde Boas).
So probably not all spaces have them...
All finite spaces have a unique minimal base. This is all U_x, where U_x = /\{U: U open and x in U},
for x in X. Note that this argument works in all spaces where all intersections of open sets are open
(sometimes called Alexandrov spaces), so e.g. it's true in discrete spaces.
I think no base for R can be minimal, e.g., so the general result for bases seems false to me.

Henno"
 
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