Automatic continuity and representation of group homomorphisms defined between groups of continuous functions

Let C(X,G) be the group of continuous functions from a topological space X into a topological group G with pointwise multiplication as the composition law, endowed with the uniform convergence topology. To what extent does the group structure of C(X,G) determine the topology of X? More generally, when does the existence of a group homomorphism H between the groups C(X,G) and C(Y,G) implies that there is a continuous map h of Y into X such that H is canonically represented by h? We prove that, for any topological group G and compact spaces X and Y, every non-vanishing C-isomorphism (defined below) H of C(X,G) into C(Y,G) is automatically continuous and can be canonically represented by a continuous map h of Y into X. Some applications to specific groups and examples are given in the paper.     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.     

Range universal spaces

We characterize those topological spaces Y for which the Isbell and finest splitting topologies on the set C(X,Y) of all continuous functions from X into Y coincide for all topological spaces X. We also consider the same question for the coincidence of the restriction of the finest splitting topology on the upper semicontinuous set-valued functions to C(X,Y) and the finest splitting topology on C(X,Y). In the first case, the spaces in question are, after identifying points that are in each others closures, subsets of the two point Sierpiński space, which gives a converse and generalization of a result of S. Dolecki, G.H. Greco, and A. Lechicki. In the second case, the spaces in question are, after identifying points that are in each others closures, order bases for bounded complete continuous DCPOs with the Scott topology.     

Integral extensions on rings of continuous functions

Let π:X→Y be a surjective continuous map between Tychonoff spaces. The map π induces, by composition, an injective morphism C(Y)→C(X) between the corresponding rings of real-valued continuous functions, and this morphism allows us to consider C(Y) as a subring of C(X). This paper deals with finiteness properties of the ring extension C(Y)⊆C(X) in relation to topological properties of the map π:X→Y. The main result says that, for X a compact subset of Rn, the extension C(Y)⊆C(X) is integral if and only if X decomposes into a finite union of closed subsets such that π is injective on each one of them.     

Some upper bounds for density of function spaces

Let Cα(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X which is closed under finite unions. We proved that the density of the space Cα(X,Y) is at most iw(X)⋅d(Y) where iw(X) denotes the i-weight of the Tychonoff space X, and d(Y) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function Ψ, and Y has a base consists of Ψ-convex subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal κ, there is a pathwise connected space Y such that π-weight of Y is κ, but Souslin number of the space Ck([0,1],Y) is κ².     

On order structure of the set of one-point Tychonoff extensions of a locally compact space

If a Tychonoff space X is dense in a Tychonoff space Y, then Y is called a Tychonoff extension of X. Two Tychonoff extensions Y₁ and Y₂ of X are said to be equivalent, if there exists a homeomorphism which keeps X pointwise fixed. This defines an equivalence relation on the class of all Tychonoff extensions of X. We identify those extensions of X which belong to the same equivalence classes. For two Tychonoff extensions Y₁ and Y₂ of X, we write Y₂?Y₁, if there exists a continuous function which keeps X pointwise fixed. This is a partial order on the set of all (equivalence classes of) Tychonoff extensions of X. If a Tychonoff extension Y of X is such that Y\X is a singleton, then Y is called a one-point extension of X. Let T(X) denote the set of all one-point extensions of X. Our purpose is to study the order structure of the partially ordered set (T(X),?). For a locally compact space X, we define an order-anti-isomorphism from T(X) onto the set of all nonempty closed subsets of βX\X. We consider various sets of one-point extensions, including the set of all one-point locally compact extensions of X, the set of all one-point Lindelöf extensions of X, the set of all one-point pseudocompact extensions of X, and the set of all one-point ?ech-complete extensions of X, among others. We study how these sets of one-point extensions are related, and investigate the relation between their order structure, and the topology of subspaces of βX\X. We find some lower bounds for cardinalities of some of these sets of one-point extensions, and in a concluding section, we show how some of our results may be applied to obtain relations between the order structure of certain subfamilies of ideals of C^∗(X), partially ordered with inclusion, and the topology of subspaces of βX\X. We leave some problems open.     

On the density of the space of continuous and uniformly continuous functions

For X a metrizable space and (Y,ρ) a metric space, with Y pathwise connected, we compute the density of (C(X,(Y,ρ)),σ)—the space of all continuous functions from X to (Y,ρ), endowed with the supremum metric σ. Also, for (X,d) a metric space and (Y,‖⋅‖) a normed space, we compute the density of (UC((X,d),(Y,ρ)),σ) (the space of all uniformly continuous functions from (X,d) to (Y,ρ), where ρ is the metric induced on Y by ‖⋅‖). We also prove that the latter result extends only partially to the case where (Y,ρ) is an arbitrary pathwise connected metric space.To carry such an investigation out, the notions of generalized compact and generalized totally bounded metric space, introduced by the author and A. Barbati in a former paper, turn out to play a crucial rôle. Moreover, we show that the first-mentioned concept provides a precise characterization of those metrizable spaces which attain their extent.     

A note on realizing homomorphisms of category algebras

The complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a complete matrix space X to the category algebra $\text{C}$(Y) of a Baire topological space Y are characterized as those σ-homomorphisms which are induced by continuous maps from dense G₈-subsets of Y into X. This result is used to deduce a series of related results in topology and measure theory (some of which are well-known). Finally a similar result for the complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a compact Hausdorff space X tothe category algebra $\text{C}$(Y) of a Baire topological space Y is proved.     

An asymmetric Ellis theorem

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, Tk, to obtain the following asymmetric Ellis theorem which applies to the example above:Whenever (X,⋅,T) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both (X,⋅,T) and (X,⋅,Tk), and inversion is a homeomorphism between (X,T) and (X,Tk).This generalizes the classical Ellis theorem, because T=Tk when (X,T) is locally compact Hausdorff.     

On the continuity of left translations in the LUC-compactification

Consider the continuity of left translations in the LUC-compactification GLUC of a locally compact group G. For every X⊆G, let κ(X) be the minimal cardinality of a compact covering of X in G. Let U(G) be the points in GLUC that are not in the closure of any X⊆G with κ(X)<κ(G). We show that the points at which no left translation in U(G) is continuous are dense in U(G). This result is a generalization of a theorem by van Douwen concerning discrete groups. We obtain a new proof for the fact that the topological center of GLUC?G is empty.     

Coincidence of the Isbell and fine Isbell topologies

Let C(X,Y) be the set of all continuous functions from a topological space X into a topological space Y. We find conditions on X that make the Isbell and fine Isbell topologies on C(X,Y) equal for all Y. For zero-dimensional spaces X, we show there is a space Z such that the coincidence of the Isbell and fine Isbell topologies on C(X,Z) implies the coincidence on C(X,Y) for all Y. We then consider the question of when the Isbell and fine Isbell topologies coincide on the set of continuous real-valued functions. Our results are similar to results established for consonant spaces.     

Topologizing homeomorphism groups of rim-compact spaces

Let X be a Tychonoff space, H(X) the group of all self-homeomorphisms of X with the usual composition and the evaluation function. Topologies on H(X) providing continuity of the evaluation function are called admissible. Topologies on H(X) compatible with the group operations are called group topologies. Whenever X is locally compact T₂, there is the minimum among all admissible group topologies on H(X). That can be described simply as a set-open topology, further agreeing with the compact-open topology if X is also locally connected. We show the same result in two essentially different cases of rim-compactness. The former one, where X is rim-compact T₂ and locally connected. The latter one, where X agrees with the rational number space Q equipped with the euclidean topology. In the first case the minimal admissible group topology on H(X) is the closed-open topology determined by all closed sets with compact boundaries contained in some component of X. Moreover, whenever X is also separable metric, it is Polish. In the rational case the minimal admissible group topology on H(Q) is just the closed-open topology. In both cases the minimal admissible group topology on H(X) is closely linked to the Freudenthal compactification of X. The Freudenthal compactification in rim-compactness plays a key role as the one-point compactification does in local compactness. In the rational case we investigate whether the fine or Whitney topology on H(Q) induces an admissible group topology on H(Q) stronger than the closed-open topology.     

Lelek maps and n-dimensional maps from compacta to polyhedra

For a natural number m?0, a map from a compactum X to a metric space Y is an m-dimensional Lelek map if the union of all non-trivial continua contained in the fibers of f is of dimension ?m. In [M. Levin, Certain finite-dimensional maps and their application to hyperspaces, Israel J. Math. 105 (1998) 257-262], Levin proved that in the space C(X,I) of all maps of an n-dimensional compactum X to the unit interval I=[0,1], almost all maps are (n−1)-dimensional Lelek maps. Moreover, he showed that in the space C(X,Ik) of all maps of an n-dimensional compactum X to the k-dimensional cube Ik (k?1), almost all maps are (n−k)-dimensional Lelek maps. In this paper, we generalize Levin's result. For any (separable) metric space Y, we define the piecewise embedding dimension ped(Y) of Y and we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a complete metric ANR Y, almost all maps are (n−k)-dimensional Lelek maps, where k=ped(Y). As a corollary, we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a Peano curve Y, almost all maps are (n−1)-dimensional Lelek maps and in the space C(X,M) of all maps of an n-dimensional compactum X to a k-dimensional Menger manifold M, almost all maps are (n−k)-dimensional Lelek maps. It is known that k-dimensional Lelek maps are k-dimensional maps for k?0.     

The notion of o-tightness and C-embedded subspaces of products

We consider the question: when is a dense subset of a space XC-embedded in X? We introduce the notion of o-tightness and prove that if each finite subproduct of a product X = Π_α?AX_α has a countable o-tightness and Y is a subset of X such that π_B(Y) = Π_α?BX_α for every countable B ? A, then Y is C-embedded in X. This result generalizes some of Noble and Ulmer's results on C-embedding.     

Almost maximal spaces

A topological space X is called almost maximal if it is without isolated points and for every x∈X, there are only finitely many ultrafilters on X converging to x. We associate with every countable regular homogeneous almost maximal space X a finite semigroup Ult(X) so that if X and Y are homeomorphic, Ult(X) and Ult(Y) are isomorphic. Semigroups Ult(X) are projectives in the category F of finite semigroups. These are bands decomposing into a certain chain of rectangular components. Under MA, for each projective S in F, there is a countable almost maximal topological group G with Ult(G) isomorphic to S. The existence of a countable almost maximal topological group cannot be established in ZFC. However, there are in ZFC countable regular homogeneous almost maximal spaces X with Ult(X) being a chain of idempotents.     

Subadditive Separating Maps

Let X and Y denote compact Hausdorff spaces and let K = R (real numbers) or C(complex numbers). C(X) and C(Y) denote the spaces of K-valued continuous functions on X and Y, respectively. A map H : C(X) C(Y) is separating if fg = 0 implies that HfHg = 0. Results about automatic continuity and the form of additive and linear separating maps have been developed in [1], [2], [3], [4], [5], [7], [8], and [10]. In this article similar results are developed for subadditive separating maps. We show (Theorem 5.11) that certain biseparating, subadditive bijections H are automatically continuous.     

Projective versions of selection principles

All spaces are assumed to be Tychonoff. A space X is called projectively P (where P is a topological property) if every continuous second countable image of X is P. Characterizations of projectively Menger spaces X in terms of continuous mappings , of Menger base property with respect to separable pseudometrics and a selection principle restricted to countable covers by cozero sets are given. If all finite powers of X are projectively Menger, then all countable subspaces of Cp(X) have countable fan tightness. The class of projectively Menger spaces contains all Menger spaces as well as all σ-pseudocompact spaces, and all spaces of cardinality less than d. Projective versions of Hurewicz, Rothberger and other selection principles satisfy properties similar to the properties of projectively Menger spaces, as well as some specific properties. Thus, X is projectively Hurewicz iff Cp(X) has the Monotonic Sequence Selection Property in the sense of Scheepers; βX is Rothberger iff X is pseudocompact and projectively Rothberger. Embeddability of the countable fan space Vω into Cp(X) or Cp(X,2) is characterized in terms of projective properties of X.     

Topological-algebraic properties of function spaces with set-open topologies

For a Tychonoff space X, we denote by Cλ(X) the space of all real-valued continuous functions on X with set-open topology. In this paper, we study the topological-algebraic properties of Cλ(X). Our main results state that (1) Cλ(X) is a topological vector space (a topological group) iff λ is a family of C-compact sets and Cλ(X)=Cλ^′(X), where λ^′ consists of all C-compact subsets of every set of λ. In particular, if Cλ(X) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family λ; (2) a topological group Cλ(X) is ω-narrow iff λ is a family of metrizable compact subsets of X.     

When is the Isbell topology a group topology?

Conditions on a topological space X under which the space C(X,R) of continuous real-valued maps with the Isbell topology κ is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in Cκ(X,R) if and only if X is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in Cκ(X,R) if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if X is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces.