The Theory of Ultrafilters

Comfort, W. W.; Negrepontis, S.

doi:10.1007/978-3-642-65780-1

Cited by 437 publications

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“…A < K. There exist II I-good countably incomplete ultrafilters on I (in fact exp2(III) of them [17]) and this is the maximal degree of goodness possible. The major lemma which will be of use to us here is the following.…”

Section: (Ie S C T => F(t) C F(s)) "Dominates" a "Multiplicative" Gmentioning

confidence: 97%

“…What we really need to show is that if <Xi: i E I> is any family of compact Hausdorff spaces such that {i: Ind(Xi) < n} E 9, then Ind(Zg, Xi) < n. (1.7) establishes the assertion for n = 0; so we work on the inductive step, and assume the more general statement true for dimensions < n. Assume Ind(X) < (ii) The inequality in (2.2.3), as well as the corresponding one in (2.2.2 Recall that for any space X, the weight w(X) of X is the least infinite cardinal of a basis for X. If X is infinite Boolean, then it is easily shown [17] that w(X) = IB(X)I. We thus immediately get the fact that if X is an infinite Boolean space and 9 is a regular ultrafilter on I then w(ZgX) = w(X)I1.…”

Section: H?g MI Where Mi Is Open (Closed) In Xi It Is Easy To Verifmentioning

confidence: 99%

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Reduced coproducts of compact Hausdorff spaces

Bankston

1987

J. symb. log.

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By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the “reduced coproduct”, which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the “ultracoproduct” can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces.

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Section: (Ie S C T => F(t) C F(s)) "Dominates" a "Multiplicative" Gmentioning

confidence: 97%

Section: H?g MI Where Mi Is Open (Closed) In Xi It Is Easy To Verifmentioning

confidence: 99%

Reduced coproducts of compact Hausdorff spaces

Bankston

1987

J. symb. log.

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“…This choice satisfies (1) and (2). Finally, if f n+1 (x n+1 ) lies in the closure of a countable set of weak P -points, fix any such countable set Y n+1 with Y n+1 ∩ {f (x l ) : l ≤ n + 1} = ∅.…”

Section: P -Points and Weak P -Points In C K (X X)mentioning

confidence: 99%

“…For (2), note that by continuity of ev (Lemma 6.4), we have ev { f n , p n : n ∈ ω} ⊆ ev({ f n , p n : n ∈ ω}) = B.…”

Section: Convergent Sequences Inmentioning

confidence: 99%

Self-maps under the compact-open topology

Lupton

Pitz

2016

Topology and its Applications

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Abstract. This paper investigates the space C k (ω * , ω * ), the space of continuous self-maps on the Stone-Čech remainder of the integers, ω * , equipped with the compact-open topology. Our main results are that• Stone-Čech extensions of injective maps on ω form a dense set of weak P -points in C k (ω * , ω * ), • it is independent of ZFC whether C k (ω * , ω * ) contains P -points, and that • C k (ω * , ω * ) is not an F -space, but contains, as ω * , no non-trivial convergent sequences.

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“…Let h: U(k)-* I be the composition of/ and g. In addition, let h: ßK-* I extend h. Take s G I arbitrarily. Then g"'({¿}) is a nonempty Gs in l/(«) and consequently has nonempty interior, [CN,14.17]. Therefore,/-1g_1({¿}) has nonempty interior (in £/(*)) and consequently we can find a subset E c k so that 0 ¥= Ë n UÍk) G f-xg-xi{s}).…”

mentioning

confidence: 99%

When 𝑈(𝜅) can be mapped onto 𝑈(𝜔)

Mill¹

1980

Proc. Amer. Math. Soc.

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U ( κ ) U(\kappa ) can be mapped onto U ( ω ) iff cf ( κ ) = ω or κ ⩾ 2 ω . U(\omega ){\text {iff}}\;{\text {cf}}(\kappa ) = \omega {\text {or}}\kappa \geqslant {2^\omega }. .

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The Theory of Ultrafilters

Cited by 437 publications

References 0 publications

Reduced coproducts of compact Hausdorff spaces

Reduced coproducts of compact Hausdorff spaces

Self-maps under the compact-open topology

When 𝑈(𝜅) can be mapped onto 𝑈(𝜔)

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