Optics in Croke-Kleiner Spaces

Fredric D. Ancel; Julia M. Wilson

Displaying similar documents to “Optics in Croke-Kleiner Spaces”

On topological groups with a small base and metrizability

Saak Gabriyelyan, Jerzy Kąkol, Arkady Leiderman (2015)

Fundamenta Mathematicae

Similarity:

A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $U_{α} : α \in ℕ^{ℕ}$ , such that $U_{α} \subset U_{β}$ whenever β ≤ α for all $α, β \in ℕ^{ℕ}$ . The class of all metrizable topological groups is a proper subclass of the class $T G_{}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G \in T G_{}$ is metrizable, and hence G is strictly angelic. We deduce from...

A note on $R_{0}$ -topological spaces

K. K. Dube (1974)

Matematički Vesnik

Similarity:

Linear combinations of partitions of unity with restricted supports

Christian Richter (2002)

Studia Mathematica

Similarity:

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 $f = \sum_{C \in} a_{C} φ_{C}$ with $a_{C} \in E$ and a partition of unity $φ_{C} : C \in$ 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...

Arhangel'skiĭ sheaf amalgamations in topological groups

Boaz Tsaban, Lyubomyr Zdomskyy (2016)

Fundamenta Mathematicae

Similarity:

We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos’s property $α_{1.5}$ is equivalent to Arhangel’skiĭ’s formally stronger property α₁. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space X such that the space $C_{p} (X)$ of continuous real-valued functions on X with the...

Nonnormality of remainders of some topological groups

Aleksander V. Arhangel&#039;skii, J. van Mill (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group $G$ has a normal remainder. In a previous paper we showed that under mild conditions on $G$ , the Continuum Hypothesis implies that if the Čech-Stone remainder $G^{*}$ of $G$ is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight...

Cellularity and the index of narrowness in topological groups

Mihail G. Tkachenko (2011)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We study relations between the cellularity and index of narrowness in topological groups and their $G_{δ}$ -modifications. We show, in particular, that the inequalities $in ({(H)}_{τ}) \leq 2^{τ \cdot in (H)}$ and $c ({(H)}_{τ}) \leq 2^{2^{τ \cdot in (H)}}$ hold for every topological group $H$ and every cardinal $τ \geq ω$ , where ${(H)}_{τ}$ denotes the underlying group $H$ endowed with the $G_{τ}$ -modification of the original topology of $H$ and $in (H)$ is the index of narrowness of the group $H$ . Also, we find some bounds for the complexity of continuous real-valued functions $f$ on an arbitrary $ω$ -narrow group...

A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

Similarity:

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $α \leq 2^{}$ , a topological group G such that $G^{γ}$ is countably compact for all cardinals γ < α, but $G^{α}$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M A_{c o u n t a b l e}$ . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M A_{c o u n t a b l e}$ . However, the question has...

On the Hausdorff Dimension of Topological Subspaces

Tomasz Szarek, Maciej Ślęczka (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

It is shown that every Polish space X with $d i m_{T} X \geq d$ admits a compact subspace Y such that $d i m_{H} Y \geq d$ where $d i m_{T}$ and $d i m_{H}$ denote the topological and Hausdorff dimensions, respectively.

Applications of some results of infinite-dimensional topology to the topological classification of operator images

Taras Banakh, Tadeusz Dobrowolski, Anatoliĭ Plichko

Similarity:

This volume consists of three relatively independent articles devoted to the topological study of the so-called operator images and weak unit balls of Banach spaces. These articles are: “The topological classification of weak unit balls of Banach spaces” by T. Banakh, “The topological and Borel classification of operator images” by T. Banakh, T. Dobrowolski and A. Plichko, and “Operator images homeomorphic to $Σ^{ω}$ ” by T. Banakh. The articles summarize investigations that has been done by...

Linear topological properties of the Lumer-Smirnov class of the polydisc

Marek Nawrocki (1992)

Studia Mathematica

Similarity:

Linear topological properties of the Lumer-Smirnov class $L N_{*} (^{n})$ of the unit polydisc $^{n}$ are studied. The topological dual and the Fréchet envelope are described. It is proved that $L N_{*} (^{n})$ has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for $L N_{*} (^{n})$ .

On star covering properties related to countable compactness and pseudocompactness

Marcelo D. Passos, Heides L. Santana, Samuel G. da Silva (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove a number of results on star covering properties which may be regarded as either generalizations or specializations of topological properties related to the ones mentioned in the title of the paper. For instance, we give a new, entirely combinatorial proof of the fact that $Ψ$ -spaces constructed from infinite almost disjoint families are not star-compact. By going a little further we conclude that if $X$ is a star-compact space within a certain class, then $X$ is neither first countable...

Remarks on flat and differential $K$ -theory

Man-Ho Ho (2014)

Annales mathématiques Blaise Pascal

Similarity:

In this note we prove some results in flat and differential $K$ -theory. The first one is a proof of the compatibility of the differential topological index and the flat topological index by a direct computation. The second one is the explicit isomorphisms between Bunke-Schick differential $K$ -theory and Freed-Lott differential $K$ -theory.

Extension of point-finite partitions of unity

Haruto Ohta, Kaori Yamazaki (2006)

Fundamenta Mathematicae

Similarity:

A subspace A of a topological space X is said to be $P^{γ}$ -embedded ( $P^{γ}$ (point-finite)-embedded) in X if every (point-finite) partition of unity α on A with |α| ≤ γ extends to a (point-finite) partition of unity on X. The main results are: (Theorem A) A subspace A of X is $P^{γ}$ (point-finite)-embedded in X iff it is $P^{γ}$ -embedded and every countable intersection B of cozero-sets in X with B ∩ A = ∅ can be separated from A by a cozero-set in X. (Theorem B) The product A × [0,1] is $P^{γ}$ (point-finite)-embedded...

Some Parity Statistics in Integer Partitions

Aubrey Blecher, Toufik Mansour, Augustine O. Munagi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We study integer partitions with respect to the classical word statistics of levels and descents subject to prescribed parity conditions. For instance, a partition with summands $λ ₁ \geq \dots \geq λ_{k}$ may be enumerated according to descents $λ_{i} > λ_{i + 1}$ while tracking the individual parities of $λ_{i}$ and $λ_{i + 1}$ . There are two types of parity levels, E = E and O = O, and four types of parity-descents, E > E, E > O, O > E and O > O, where E and O represent arbitrary even and odd summands. We obtain functional equations...

The (dis)connectedness of products of Hausdorff spaces in the box topology

Vitalij A. Chatyrko (2021)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In this paper the following two propositions are proved: (a) If $X_{α}$ , $α \in A$ , is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product $□_{α \in A} X_{α}$ can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If $X_{α}$ , $α \in A$ , is an infinite system of Brown Hausdorff topological spaces then the box product $□_{α \in A} X_{α}$ is also Brown Hausdorff, and hence, it is connected. A space...

Remarks on $\mathcal{S}$ -Closedness in Topological Spaces

Zbigniew Duszyński (2007)

Bollettino dell'Unione Matematica Italiana

Similarity:

Corresponding to [27], some properties of S-closed subspaces and subsets $\mathcal{S}$ -closed relative to a topological space are proved. Conditions under which mappings preserve certain $\mathcal{S}$ -closed subspaces are investigated.