Remarks on flat and differential $K$-theory

Man-Ho Ho

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A note on $R_{0}$ and $R_{1}$ topological spaces

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A countably cellular topological group all of whose countable subsets are closed need not be $ℝ$ -factorizable

Mihail G. Tkachenko (2023)

Commentationes Mathematicae Universitatis Carolinae

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We construct a Hausdorff topological group $G$ such that $ℵ_{1}$ is a precalibre of $G$ (hence, $G$ has countable cellularity), all countable subsets of $G$ are closed and $C$ -embedded in $G$ , but $G$ is not $ℝ$ -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.

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We define combinatorial structures which we refer to as flat morasses, and use them to construct a Lindelöf space with points $G_{δ}$ of cardinality $ℵ_{ω}$ , consistent with GCH. The construction reveals, it is hoped, that flat morasses are a tool worth adding to the kit of any user of set theory.

Nonnormality of remainders of some topological groups

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

Commentationes Mathematicae Universitatis Carolinae

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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...

On $n$ -thin dense sets in powers of topological spaces

Adam Bartoš (2016)

Commentationes Mathematicae Universitatis Carolinae

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A subset of a product of topological spaces is called $n$ -thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_{3}$ space $X$ without isolated points such that $X^{n}$ contains an $n$ -thin dense subset, but $X^{n + 1}$ does not contain any $n$ -thin dense subset. We also observe that part of the construction can be carried out under MA.

Characterizations of $z$ -Lindelöf spaces

Ahmad Al-Omari, Takashi Noiri (2017)

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A topological space $(X, τ)$ is said to be $z$ -Lindelöf [1] if every cover of $X$ by cozero sets of $(X, τ)$ admits a countable subcover. In this paper, we obtain new characterizations and preservation theorems of $z$ -Lindelöf spaces.

$C^{*}$ -points vs $P$ -points and $P^{♭}$ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space $X$ , the point $p \in X$ is called a $C^{*}$ -point if every real-valued continuous function on $C ∖ {p}$ can be extended continuously to $p$ . Every point in an extremally disconnected space is a $C^{*}$ -point. A classic example is the space $𝐖^{*} = ω_{1} + 1$ consisting of the countable ordinals together with $ω_{1}$ . The point $ω_{1}$ is known to be a $C^{*}$ -point as well as a $P$ -point. We supply a characterization of $C^{*}$ -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

An irrational problem

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Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define $X_{M}$ to be X ∩ M with topology generated by $U \cap M : U \in \cap M$ . Suppose $X_{M}$ is homeomorphic to the irrationals; must $X = X_{M}$ ? We have partial results. We also answer a question of Gruenhage by showing that if $X_{M}$ is homeomorphic to the “Long Cantor Set”, then $X = X_{M}$ .

Algebraic and topological properties of some sets in ℓ₁

Taras Banakh, Artur Bartoszewicz, Szymon Głąb, Emilia Szymonik (2012)

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For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series $\sum_{n = 1}^{\infty} x (n)$ . Guthrie and Nymann proved that E(x) is one of the following types of sets: () a finite union of closed intervals; () homeomorphic to the Cantor set; homeomorphic to the set T of subsums of $\sum_{n = 1}^{\infty} b (n)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable....

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

Fixed points with respect to the L-slice homomorphism $σ_{a}$

K.S. Sabna, N.R. Mangalambal (2019)

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Given a locale $L$ and a join semilattice $J$ with bottom element $0_{J}$ , a new concept $(σ, J)$ called $L$ -slice is defined,where $σ$ is as an action of the locale $L$ on the join semilattice $J$ . The $L$ -slice $(σ, J)$ adopts topological properties of the locale $L$ through the action $σ$ . It is shown that for each $a \in L$ , $σ_{a}$ is an interior operator on $(σ, J)$ .The collection $M = {σ_{a}; a \in L}$ is a Priestly space and a subslice of $L$ - $Hom (J, J)$ . If the locale $L$ is spatial we establish an isomorphism between the $L$ -slices $(σ, L)$ and $(δ, M)$ . We have shown that the fixed...

Generalized versions of Ilmanen lemma: Insertion of $C^{1, ω}$ or $C_{loc}^{1, ω}$ functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a normed linear space $X$ , if $f_{1} : X \to ℝ$ is continuous and semiconvex with modulus $ω$ , $f_{2} : X \to ℝ$ is continuous and semiconcave with modulus $ω$ and $f_{1} \leq f_{2}$ , then there exists $f \in C^{1, ω} (X)$ such that $f_{1} \leq f \leq f_{2}$ . Using this result we prove a generalization of Ilmanen lemma (which deals with the case $ω (t) = t$ ) to the case of an arbitrary nontrivial modulus $ω$ . This generalization (where a $C_{l o c}^{1, ω}$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

Displaying similar documents to “Remarks on flat and differential K -theory”

Displaying similar documents to “Remarks on flat and differential $K$ -theory”