Supercompact minus compact is super

On the existence of continuous (approximate) roots of algebraic equations

A categorical characterization of a subcategory S of TOP (or 7J) is one which enables the identification of S in TOP (or 7^) without requiring the reconstruction of the topological structure of its objects.In this paper we so characterize various familiar subcategories of TOP (Hausdorff spaces, normal spaces, compact Hausdorff spaces, paracompact Hausdorff spaces, metrizable spaces, first countable spaces) in terms of the global behavior of the (objects and) morphisms of the subcategory. Introduction. A categorical characterization of a subcategory S of TOP (or of T2) is one which enables the identification of S in TOP (or T2) without requiring the reconstruction of the topological structure of its objects.The proof of the triviality of the automorphism class group of TOP [Fe, p. 32] assures us of the existence of such characterizations.As will be shown in §1, this proof establishes that all the topological structure of an object of TOP can be recovered from the behavior of the arrows with that object as source and/ or target.In §2 we establish a number of lemmas (mostly known), and we use these lemmas in §3 to characterize many important subcategories of TOP and/ or T2, such as the metrizable spaces, the paracompact Hausdorff spaces, the normal spaces, etc.We can distinguish at least three classes of "categorical" characterizations of subcategories of TOP, which are listed below: Class 3. Straightforward translations of the familiar definitions.These can always be obtained because of the results in §1.Example: Characterize the concepts "open", "cover", "refinement", "locally finite" categorically and then the standard definition of paracompact spaces is categorical.Class 2. Characterizations using standard categorical constructions (product, coproduct, epimorphic images, etc.) but in terms of individual objects, and/or constants outside the subcategory.Example: The category of paracompact Hausdorff spaces contains precisely those Hausdorff spaces X such that

Let X, Y be compact Hausdorff spaces and B£(X), B*(Y)f 0 < a, |3 < 12 (the first uncountable ordinal), the associated Banach spaces of bounded real-valued Baire functions of classes a and 0. If B*(X) =£ B*AX) (which is the case if a =£ 0 and X is not dispersed), then B*(X) is neither linearly isometric to BQ(Y) nor equivalent to B^(Y) in several other ways. B^(X) is linearly isometric to B%(Y) if and only if X is Baire isomorphic to Y. For 1 < a < ft the maximal ideal space of B£(X) for a nondispersed compact space X As not an F-space. 1. Let I bea compact (more generally, completely regular) Hausdorff space and C(X) the space of continuous real-valued functions on X. Let B0(X) = C(X)9 and inductively define Ba(X) for each ordinal a < Q, (Q denotes the first uncountable ordinal) to be the space of pointwise limits of sequences of functions in U | < a B^(X). Let B*(X) be the space of bounded functions contained in Ba(X). With the pointwise operations Ba(X) and B*(X) are lattice-ordered algebras. With the supremum norm B*(X) is a Banach algebra (see [4, §41]). The Baire sets of X of multiplicative class a, denoted by Za(X)9 are defined to be the zero sets of functions in B*(X). Those of additive class a, denoted by CZa(X), are defined as the complements of sets in Za(X). Finally, those of ambiguous class a, denoted by Aa(X), are the sets which are simultaneously in Za(X) and CZa(X). With the set-theoretic operations of union and intersection, Aa(X) is a Boolean algebra for each a < 12. The sets of exactly ambiguous class a, denoted by EAa(X are those in ^ a ( Z ) U ^ < a Ac(X). The sets of exactly additive and exactly multiplicative class a are defined analogously. The class of all Baire subsets of X is AMS (MOS) subject classifications (1970). Primary 06A65, 26A21, 28A05, 46E1S, 46E30, 46J10, 54C50, 54H05; Secondary 04A15.

Let π : Y → X \pi : Y\rightarrow X be a continuous surjection between compact Hausdorff spaces Y Y and X X which is irreducible in the sense that if F ⊊ Y F\subsetneq Y is closed, then π ( F ) ≠ X \pi (F)\neq X . We exhibit isomorphisms between various Boolean algebras associated to this data: the regular open sets of X X , the regular open sets of Y Y , the regular ideals of C ( X ) C(X) and the regular ideals of C ( Y ) C(Y) . We call X X and Y Y Boolean equivalent if the regular open sets of X X and the regular open sets of Y Y are isomorphic Boolean algebras. We give a characterization of when two compact metrizable spaces are Boolean equivalent; this characterization may be viewed as a topological version of the characterization of standard Borel spaces.

d-20 - H-Closed Spaces

In 1975 E. K. van Douwen showed that if ( X n ) n ∈ ω ( X_n )_{ n \in \omega } is a family of Hausdorff spaces such that all finite subproducts ∏ n &gt; m X n \prod _{ n &gt; m } X_n are paracompact, then for each element x x of the box product ◻ n ∈ ω X n \square _{n \in \omega } X_n the σ \sigma -product σ ( x ) = { y ∈ ◻ n ∈ ω X n : { n ∈ ω : x ( n ) ≠ y ( n ) } is finite } \sigma ( x ) = \{ y \in \square _{n \in \omega } X_n : \{ n \in \omega : x (n) \neq y (n) \} \text { is finite} \} is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result: Theorem Let κ \kappa be an infinite cardinal number, and let ( X α ) α ∈ κ ( X_{\alpha } )_{\alpha \in \kappa } be a family of compact Hausdorff spaces. Let x ∈ ◻ = ◻ α ∈ κ X α x \in \square = \square _{\alpha \in \kappa } X_\alpha be a fixed point. Given a family R \mathcal {R} of open subsets of ◻ \square which covers σ ( x ) \sigma ( x ) , there exists an open locally finite in ◻ \square refinement S \mathcal {S} of R \mathcal {R} which covers σ ( x ) \sigma ( x ) . We also prove a slightly weaker version of this theorem for Hausdorff spaces with “all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen’s question.

Function-space compactifications of function spaces

We find conditions which guarantee that a given flow Φ on a compact metric space X admits a Lyapunov 1-form ω lying in a prescribed Cech cohomology class ξ∈H ^ 1 (X;ℝ). These conditions are formulated in terms of the restriction of ξ to the chain recurrent set of Φ. The result of the paper may be viewed as a generalization of a well-known theorem by Conley about the existence of Lyapunov functions.

The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : $\dim (A\cup B)\le \dim A+\dim B+1$. Theorem. Suppose $A,B$ are subsets of a metrizable space and $K$ and $L$ are CW complexes. If $K$ is an absolute extensor for $A$ and $L$ is an absolute extensor for $B$, then the join $K*L$ is an absolute extensor for $A\cup B$. As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose $A$, $B$ are subsets of a metrizable space. Then \begin{equation*}\dim _{\mathbf {R} }(A\cup B)\le \dim _{\mathbf {R} }A+\dim _{\mathbf {R} }B+1 \end{equation*} for any ring $\mathbf {R}$ with unity and \begin{equation*}\dim _{G}(A\cup B)\le \dim _{G}A+\dim _{G}B+2\end{equation*} for any abelian group $G$. The second part of the paper is devoted to the question of existence of universal spaces: Suppose $\{K_{i}\}_{i\ge 1}$ is a sequence of CW complexes homotopy dominated by finite CW complexes. Then [a.] Given a separable, metrizable space $Y$ such that $K_{i}\in AE(Y)$, $i\ge 1$, there exists a metrizable compactification $c(Y)$ of $Y$ such that $K_{i}\in AE(c(Y))$, $i\ge 1$. [b.] There is a universal space of the class of all compact metrizable spaces $Y$ such that $K_{i}\in AE(Y)$ for all $i\ge 1$. [c.] There is a completely metrizable and separable space $Z$ such that $K_{i}\in AE(Z)$ for all $i\ge 1$ with the property that any completely metrizable and separable space $Z’$ with $K_{i}\in AE(Z’)$ for all $i\ge 1$ embeds in $Z$ as a closed subset.

Let K be a Hausdorff space and Cb(K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gâteaux and Frechet differentiability of subspaces of Cb(K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of Cb(K) is a dense Gδ subset of H, if K is compact. This gives a generalized Bishop’s theorem, which says that the closure of the set of all strong peak points for H is the smallest closed norming subset of H. The classical Bishop’s theorem was proved for a separating subalgebra H and a metrizable compact space K.

We show that the K-theory cosheaf is a complete invariant for separable continuous fields with vanishing boundary maps over a finite-dimensional compact metrizable topological space whose fibers are stable Kirchberg algebras with rational K-theory groups satisfying the universal coefficient theorem. We provide a range result for fields in this class with finite-dimensional K-theory. There are versions of both results for unital continuous fields.

A generalization of Edelstein’s version of a theorem of Janos and its converse are obtained: Theorem. Let X be a compact metrizable topological space, and f, g be continuous self-maps of X such that g f = f g gf = fg and f is bijective. Then g is injective and ⋂ 1 ∞ g n X = { x 0 } \bigcap \nolimits _1^\infty {{g^n}X = \{ {x_0}\} } , where x 0 ∈ X {x_0} \in X , iff, given λ , 0 &gt; λ &gt; 1 \lambda ,0 &gt; \lambda &gt; 1 , a homeomorphism h of X into l 2 {l_2} exists such that \[ ‖ h ( g x ) − h ( g y ) ‖ = λ ‖ h ( f x ) − h ( f y ) ‖ \left \| {h(gx) - h(gy)} \right \| = \lambda \left \| {h(fx) - h(fy)} \right \| \] for all x , y ∈ X x,y \in X .

Given two topologies J1, J2 on a set X, J1 is said to be coarser than J2, written J1 ≦ J2, if every set open under J1 is open under J2. A minimal Hausdorff space is then one for which there is no coarser Hausdorff topology etc. Vaidyanathaswamy [4] showed that every compact Hausdorff space is both maximal compact and minimal Hausdorff. This raised the question of whether there exist minimal Hausdorff non-compact spaces and/or maximal compact non-Hausdorff spaces. These questions were in fact answered in the affirmative by Ramanathan [2], Balachandran [1], and Hing Tong [3]. Their examples were, however, all on countable sets, and the topology constructed to answer one question bore no relation to the topology answering the second. In particular, the minimal Hausdorff non-compact topologies were not finer than any maximal compact topology.

By the Riesz Representation Theorem for locally compact Hausdorff spaces, for every positive linear functional I on K(X) there is a measure @m such that I(f)=@!fd@m, where K(X) is the set of continuous real functions with compact support on the locally compact Hausdorff space X. In this article we prove a uniformly computable version of this theorem for computably locally compact computable Hausdorff spaces X. We introduce a representation of the positive linear functionals I on K(X) and a representation of the Borel measures on X and prove that for every such functional I a measure @m can be computed and vice versa such that I(f)=@!fd@m.

Let G be a locally compact second countable Hausdorff space with a positive regular Borel measure λ, where λ is invariant under a continuous injective mapping φ : G → G. We characterize the disjoint hypercyclicity of finite weighted translations generated by φ acting on the weighted space Lp (G, ω) (1 ≤ p &lt; ∞).