arXiv subjects

S. Dolecki

Publications and source records attributed to S. Dolecki.

3 recordsLinked to original sources

arXiv2010

Group topologies coarser than the Isbell topology

The Isbell, compact-open and point-open topologies on the set $C(X,\mathbb{R})$ of continuous real-valued maps can be represented as the dual topologies with respect to some collections $α(X)$ of compact families of open subsets of a topological space $X$. Those $α(X)$ for which addition is jointly continuous at the zero function in $C_α(X,\mathbb{R})$ are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections $α(X)$ for which $C_α(X,\mathbb{R})$ is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if $X$ is infraconsonant. Examples based on measure theoretic methods, that $C_α(X,\mathbb{R})$ can be strictly finer than the compact-open topology, are given. To our knowledge, this is the first example of a splitting group topology strictly finer than the compact-open topology.

math.GN↗

arXiv2010

A unified theory of function spaces and hyperspaces: local properties

Many classically used function space structures (including the topology of pointwise convergence, the compact-open topology, the Isbell topology and the continuous convergence) are induced by a hyperspace structure counterpart. This scheme is used to study local properties of function space structures on $C(X,\mathbb R)$, such as character, tighntess, fan-tightness, strong fan-tightness, the Fr{é}chet property and some of its variants. Under mild conditions, local properties of $C(X,\mathbb R)$ at the zero function correspond to the same property of the associated hyperspace structure at $X$. The latter is often easy to characterize in terms of covering properties of $X$. This way, many classical results are recovered or refined, and new results are obtained. In particular, it is shown that tightness and character coincide for the continuous convergence on $C(X,\mathbb R)$ and is equal to the Lindel{ö}f degree of $X$. As a consequence, if $X$ is consonant, the tightness of $C(X,\mathbb R)$ for the compact-open topology is equal to the Lindel{ö}f degree of $X$.

math.GN↗

arXiv2010

When is the {I}sbell topology a group topology?

Conditions on a topological space $X$ under which the space $C(X,\mathbb{R})$ 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 $C_κ(X,\mathbb{R})$ 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 $C_κ(X,\mathbb{R})$ 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.

math.GN↗