Monotonically Normal Research Articles

Dual properties and monotone normality

Dual properties and monotone normality

Monotone normality and related proprieties

In this paper monotone versions of some results on normality and on property (a) are investigated.

Monotone normality in generalized topological spaces

Monotone normality in generalized topological spaces is introduced. The characterizations and several preservation theorems of μ-monotonically normal spaces are given and the “monotone variant” of Urysohn’s lemma for μ-monotonically normal spaces is established.

An application of Mary Ellen Rudin's solution to Nikiel's Conjecture

The concept of utter normality is a formal strengthening of the concept of monotone normality. It is an unsolved problem whether every monotonically normal space is utterly normal. Here we show that every suborderable monotonically normal space and every compact monotonically normal space is utterly normal. The latter result makes use of Mary Ellen Rudin's powerful theorem that every compact, monotonically normal space is the continuous image of a compact linearly ordered space. Corollaries include the fact that a compact, monotonically normal space is hereditarily weakly orthocompact. We show that a locally compact, monotonically normal space has a monotonically normal compactification if, and only if, it is weakly orthocompact. Mary Ellen Rudin's example [13] of a locally compact, monotonically normal space whose one point compactification is not monotonically normal shows that some such characterization is necessary.

Mary Ellen Rudin and monotone normality

This paper focuses on the remarkable contributions that Mary Ellen Rudin made to the study of monotonically normal spaces.

Separation of diagonal in monotonically normal spaces and their products

As separation of diagonal, we study when monotone normality implies Δ-paracompactness or Δ-normality. For that, it is proved that every monotonically normal space is Δ-paracompact if the projection of its square is closed. Moreover, it is proved that every monotonically normal space is Δ-normal if it has countable tightness (or countable extent). In particular, the parenthetic part is an affirmative answer to Burke and Buzyakova's problem in 2010. Secondly, we study the relation between normality and Δ-paracompactness or Δ-normality in certain products. For that, we additionally introduce two new neighborhood properties. Using these ones, it is proved that the product X×K of a monotonically normal space X and a compact space K is Δ-paracompact (respectively, Δ-normal) if and only if X is Δ-paracompact (respectively, Δ-normal) and X×K is normal.

Monotone normality and stratifiability from a pointfree point of view

Monotone normality is usually defined in the class of T1 spaces. In this paper we study it under the weaker condition of subfitness, a separation condition that originates in pointfree topology. In particular, we extend some well known characterizations of these spaces to the subfit context (notably, their hereditary property and the preservation under surjective continuous closed maps) and present a similar study for stratifiable spaces, an important subclass of monotonically normal spaces. In the second part of the paper, we extend further these ideas to the lattice theoretic setting. In particular, we give the pointfree analogues of the previous results on monotonically normal spaces and introduce and investigate the natural pointfree counterpart of stratifiable spaces.

Monotone normality and neighborhood assignments

We prove that every neighborhood assignment for a monotonically normal space has a kernel which is homeomorphic to some subspace of an ordinal. As a corollary, every monotone neighborhood assignment for a monotonically normal space has a discrete kernel, which gives a partial answer to a question posed in Buzyakova et al. (2007) [5]. We also give an example of a regular space which has a neighborhood assignment with no kernels homeomorphic to any subspace of an ordinal.

Quasi-metrics and monotone normality

The purpose of this paper is to study which quasi-metrizable spaces are monotonically normal. In particular, we provide a sufficient condition for a quasi-metrizable space to be monotonically normal. This enables us to prove the monotone normality of a certain amount of interesting examples of quasi-metric spaces; for instance, we show that the continuous poset of formal balls of a metric space, endowed with the Scott topology, is a monotonically normal quasi-metrizable space.

Monotone versions of δ-normality

According to Mack a space is countably paracompact if and only if its product with [ 0 , 1 ] is δ-normal, i.e. any two disjoint closed sets, one of which is a regular G δ -set, can be separated. In studying monotone versions of countable paracompactness, one is naturally led to consider various monotone versions of δ-normality. Such properties are the subject of this paper. We look at how these properties relate to each other and prove a number of results about them, in particular, we provide a factorization of monotone normality in terms of monotone δ-normality and a weak property that holds in monotonically normal spaces and in first countable Tychonoff spaces. We also discuss the productivity of these properties with a compact metrizable space.

On Monotonically T2-spaces and Monotonicallynormal spaces

In this paper we show that if ? Xi is monotonically T2-space then each Xi is monotonically T2-space, too. Moreover, we show that if ? Xi is monotonically normal space then each Xi is monotonically normal space, too. Among these results we give a new proof to show that the monotonically T2-space property and monotonically normal space property are hereditary property and topologically property and give an example of T2-space but not monotonically T2-space.

On Monotonically T2-spaces and Monotonicallynormal spaces

In this paper we show that if ? Xi is monotonically T2-space then each Xi is monotonically T2-space, too. Moreover, we show that if ? Xi is monotonically normal space then each Xi is monotonically normal space, too. Among these results we give a new proof to show that the monotonically T2-space property and monotonically normal space property are hereditary property and topologically property and give an example of T2-space but not monotonically T2-space.

Resolvability and monotone normality

A space X is said to be κ-resolvable (resp., almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp., almost disjoint over the ideal of nowhere dense subsets). X is maximally resolvable if and only if it is Δ(X)-resolvable, where Δ(X) = min{|G| : G ≠ $$\not 0$$ open}. We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost μ-resolvable, where μ = min{2 ω , ω 2}. On the other hand, if κ is a measurable cardinal then there is a MN space X with Δ(X) = κ such that no subspace of X is ω 1-resolvable. Any MN space of cardinality < ℵ ω is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = Δ(X) = ℵ ω such that no subspace of X is ω 2-resolvable.

Monotone normality free of T 1 axiom

Monotone normality is usually defined in the class of T1 spaces. In this paper new characterizations of monotone normality, free of T1 axiom, are provided and it is shown that in this context it is not a hereditary property. Also, a Tietze-type extension theorem for lattice-valued functions for this class of spaces is given.

Monotone insertion of lattice-valued functions

Insertion of lattice-valued functions in a monotone manner is investigated. For L a ⊲-separable completely distributive lattice (i.e. L admits a countable base which is free of supercompact elements), a monotone version of the Katětov-Tong insertion theorem for L-valued functions is established. We also provide a monotone lattice-valued version of Urysohn’s lemma. Both results yield new characterizations of monotonically normal spaces. Moreover, extension of lattice-valued functions under additional assumptions is shown to characterize also monotone normality.

REGULAR RELATIONS AND MONOTONE NORMAL ORDERED SPACES

In this paper the classical theorem of Zareckiĭ about regular relations is generalized and an intrinsic characterization of regularity is obtained. Based on the generalized Zareckiĭ theorem and the intrinsic characterization of regularity, the authors give a characterization of monotone normality of ordered spaces. A new proof of the Urysohn-Nachbin lemma is presented which is quite different from the classical one.

A Cyclic Element Characterization of Monotone Normality

A Cyclic Element Characterization of Monotone Normality

Metrizability, monotone normality, and other strong properties in trees

Metrizability is an extremely strong property where trees are concerned, and it turns out that in many ways, monotone normality is the appropriate generalization when the trees have uncountable chains. We show that monotone normality is equivalent to the tree being the topological direct sum of ordinal spaces, each of which is a convex chain in the tree. Several metrization theorems are proven, some in ZFC, some just assuming ZF or “ZF + Countable AC”, and still others assuming ZFC-independent axioms, as well as theorems in a similar spirit with monotone normality of the tree as a conclusion. The property of being collectionwise Hausdorff plays a key role, and we obtain partial results on the still unsolved problems of whether it is consistent that every collectionwise Hausdorff tree or every normal tree is monotone normal.

Monotone normality in products

Monotone normality in finite and infinite topological products is investigated. As shown in (Heath et al., 1973), the countable (Tychonoff) power of a space is monotonically normal if and only if the space is stratifiable. It is shown that if the square of a space is monotonically normal, then all finite powers are monotonically normal and hereditarily paracompact. For certain special cases, it is observed that a space has all finite powers monotonically normal if and only if it linearly stratifiable. Nonetheless, a monotonically normal topological group is constructed, all of whose finite powers are monotonically normal, but which is not linearly stratifiable. The group is constructed using special filters and nonstandard topologies on infinite products.

Topological properties of spaces ordered by preferences

In this paper, we analyze the main topological properties of a relevant class of topologies associated with spaces ordered by preferences (asymmetric, negatively transitive binary relations). This class consists of certain continuous topologies which include the order topology. The concept of saturated identification is introduced in order to provide a natural proof of the fact that all these spaces possess topological properties analogous to those of linearly ordered topological spaces, inter alia monotone and hereditary normality, and complete regularity.