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Intersection topologies with respect to separable GO-spaces and the countable ordinals

M. Jones (1995)

Fundamenta Mathematicae

Given two topologies, T 1 and T 2 , on the same set X, the intersection topologywith respect to T 1 and T 2 is the topology with basis U 1 ∩ U 2 : U 1 ∈ T 1 , U 2 ∈ T 2 . Equivalently, T is the join of T 1 and T 2 in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and ω 1 -compactness in this class of topologies. We demonstrate that the majority of his results generalise...

Large cardinals and Dowker products

Chris Good (1994)

Commentationes Mathematicae Universitatis Carolinae

We prove that if there is a model of set-theory which contains no first countable, locally compact, scattered, countably paracompact space X , whose Tychonoff square is a Dowker space, then there is an inner model which contains a measurable cardinal.

Linear combinations of partitions of unity with restricted supports

Christian Richter (2002)

Studia Mathematica

Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation f = ∑ C ∈ a C φ C with a C ∈ E and a partition of unity φ C : C ∈ subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction f | P attains its extrema at vertices of P. Finally, a class of extremal functions on the metric space...

Locally compact perfectly normal spaces may all be paracompact

Paul B. Larson, Franklin D. Tall (2010)

Fundamenta Mathematicae

We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space...

Monotone normality and extension of functions

Ian Stares (1995)

Commentationes Mathematicae Universitatis Carolinae

We provide a characterisation of monotone normality with an analogue of the Tietze-Urysohn theorem for monotonically normal spaces as well as answer a question due to San-ou concerning the extension of Urysohn functions in monotonically normal spaces. We also extend a result of van Douwen, giving a characterisation of K 0 -spaces in terms of semi-continuous functions, as well as answer another question of San-ou concerning semi-continuous Urysohn functions.

Monotonically normal e -separable spaces may not be perfect

John E. Porter (2018)

Commentationes Mathematicae Universitatis Carolinae

A topological space X is said to be e -separable if X has a σ -closed-discrete dense subset. Recently, G. Gruenhage and D. Lutzer showed that e -separable PIGO spaces are perfect and asked if e -separable monotonically normal spaces are perfect in general. The main purpose of this article is to provide examples of e -separable monotonically normal spaces which are not perfect. Extremely normal e -separable spaces are shown to be stratifiable.

New proofs of classical insertion theorems

Chris Good, Ian Stares (2000)

Commentationes Mathematicae Universitatis Carolinae

We provide new proofs for the classical insertion theorems of Dowker and Michael. The proofs are geometric in nature and highlight the connection with the preservation of normality in products. Both proofs follow directly from the Katětov-Tong insertion theorem and we also discuss a proof of this.

Non-normality points and nice spaces

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

J. Terasawa in " β X - { p } are non-normal for non-discrete spaces X " (2007) and the author in “On non-normality points and metrizable crowded spaces” (2007), independently showed for any metrizable crowded space X that each point p of its Čech–Stone remainder X * is a non-normality point of β X . We introduce a new class of spaces, named nice spaces, which contains both of Sorgenfrey line and every metrizable crowded space. We obtain the result above for every nice space.

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