Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, Dec 7, 2012
The operation of extending functions from $$\scriptstyle X$$ to $$\scriptstyle \upsilon X$$ is $$... more The operation of extending functions from $$\scriptstyle X$$ to $$\scriptstyle \upsilon X$$ is $$\scriptstyle \omega $$-continuous, so it is natural to study $$\scriptstyle \omega $$-continuous maps systematically if we want to find out which properties of $$\scriptstyle C_p(X)$$ “lift” to $$\scriptstyle C_p(\upsilon X)$$. We study the properties preserved by $$\scriptstyle \omega $$-continuous maps and bijections both in general spaces and in $$\scriptstyle C_p(X)$$. We show that $$\scriptstyle \omega $$-continuous maps preserve primary $$\scriptstyle \Sigma $$-property as well as countable compactness. On the other hand, existence of an $$\scriptstyle \omega $$-continuous injection of a space $$\scriptstyle X$$ to a second countable space does not imply $$\scriptstyle G_\delta $$-diagonal in $$\scriptstyle X$$; however, existence of such an injection for a countably compact $$\scriptstyle X$$ implies metrizability of $$\scriptstyle X$$. We also establish that $$\scriptstyle \omega $$-continuous injections can destroy caliber $$\scriptstyle \omega _1$$ in pseudocompact spaces. In the context of relating the properties of $$\scriptstyle C_p(X)$$ and $$\scriptstyle C_p(\upsilon X)$$, a countably compact subspace of $$\scriptstyle C_p(X)$$ remains countably compact in the topology of $$\scriptstyle C_p(\upsilon X)$$; however, compactness, pseudocompactness, Lindelöf property and Lindelöf $$\scriptstyle \Sigma $$-property can be destroyed by strengthening the topology of $$\scriptstyle C_p(X)$$ to obtain the space $$\scriptstyle C_p(\upsilon X)$$. We show that Lindelöf$$\scriptstyle \Sigma $$-property of $$\scriptstyle C_p(X)$$ together with $$\scriptstyle \omega _1$$ being a caliber of $$\scriptstyle C_p(X)$$ implies that $$\scriptstyle X$$ is cosmic.
Commentationes Mathematicae Universitatis Carolinae, 2009
We study relations between the Lindelof property in the spaces of continuous functions with the t... more We study relations between the Lindelof property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if $C_p(X)$ is Lindelof, $Y=X\cup\{p\}$, and the point $p$ has countable character in $Y$, then $C_p(Y)$ is Lindelof; b) if $Y$ is a cozero subspace of a Tychonoff space $X$, then $l(C_p(Y)^\omega)\le l(C_p(X)^\omega)$ and $\operatorname{ext}(C_p(Y)^\omega)\le \operatorname{ext}(C_p(X)^\omega)$.
Abstract We prove that t m ( X × Y ) ≤ t m ( X ) t m ( Y ) if the space Y is locally compact, and... more Abstract We prove that t m ( X × Y ) ≤ t m ( X ) t m ( Y ) if the space Y is locally compact, and that always t m ( X × Y ) ≤ t m ( X ) χ ( Y ) , where t m ( Z ) is the minitightness (a.k.a. the weak functional tightness) of a space Z .
We prove, modifying Dowker’s example, that there exists a normal space X such that is Lindelof, X... more We prove, modifying Dowker’s example, that there exists a normal space X such that is Lindelof, X is zero-dimensional and is not strongly zero-dimensional.
This chapter provides an overview of tightness and t -equivalence. Two spaces X and Y are called ... more This chapter provides an overview of tightness and t -equivalence. Two spaces X and Y are called M- equivalent if their free topological groups F ( X ) and F ( Y ) in the sense of Markov are topologically isomorphic. The spaces X and Y are l- equivalent if the spaces C p ( X ) and C p ( Y ) of real-valued continuous functions equipped with the topology of point-wise convergence are linearly homeomorphic, and t- equivalent if C p ( X ) and C p ( Y ) are homeomorphic; Arhangel'skiĭ has shown that M -equivalence of two spaces implies their l -equivalence; clearly, l- equivalent spaces are t -equivalent. A topological property is preserved by an equivalence relation if whenever two spaces are in the relation, one of them has the property if and only if the other one does. Similarly, a cardinal invariant is preserved by a relation if its values on two spaces are the same whenever the spaces are in the relation. The chapter discusses about the example that shows the non-preservation of the tightness by M -equivalence depends heavily on the fact that one of the two spaces is not normal. The chapter also discuses problems based on M -equivalent Lindelof spaces.
This chapter provides an overview of LΣ ( κ )-spaces. If κ be a cardinal (finite or infinite), th... more This chapter provides an overview of LΣ ( κ )-spaces. If κ be a cardinal (finite or infinite), then a space X is an LΣ (≤ κ )- space if there is a second-countable space M and a compact-valued upper semi-continuous mapping p : M → X such that p ( M ) = X and w ( p ( x )) ≤ κ for every x ∈ X ( w (·) denotes weight). A space X is an LΣ ( κ )-space if it is an LΣ (≤)‑space and is not an LΣ (≤ λ )-space for any λ κ. A space X is an LΣ ( κ )-space if there is a second-countable space M and a compact-valued upper semi-continuous mapping p : M → X such that p ( M ) = X and w ( p ( x )) κ for every x ∈ X. For finite k the definition says that the images of points under p have at most κ points. The class LΣ (<ω) is the class of all images of second-countable spaces under finite-valued upper semicontinuous mappings. The definitions also admit natural reformulations in terms of networks modulo compact covers in the spirit of the seminal article of K. Nagami.
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, Dec 7, 2012
The operation of extending functions from $$\scriptstyle X$$ to $$\scriptstyle \upsilon X$$ is $$... more The operation of extending functions from $$\scriptstyle X$$ to $$\scriptstyle \upsilon X$$ is $$\scriptstyle \omega $$-continuous, so it is natural to study $$\scriptstyle \omega $$-continuous maps systematically if we want to find out which properties of $$\scriptstyle C_p(X)$$ “lift” to $$\scriptstyle C_p(\upsilon X)$$. We study the properties preserved by $$\scriptstyle \omega $$-continuous maps and bijections both in general spaces and in $$\scriptstyle C_p(X)$$. We show that $$\scriptstyle \omega $$-continuous maps preserve primary $$\scriptstyle \Sigma $$-property as well as countable compactness. On the other hand, existence of an $$\scriptstyle \omega $$-continuous injection of a space $$\scriptstyle X$$ to a second countable space does not imply $$\scriptstyle G_\delta $$-diagonal in $$\scriptstyle X$$; however, existence of such an injection for a countably compact $$\scriptstyle X$$ implies metrizability of $$\scriptstyle X$$. We also establish that $$\scriptstyle \omega $$-continuous injections can destroy caliber $$\scriptstyle \omega _1$$ in pseudocompact spaces. In the context of relating the properties of $$\scriptstyle C_p(X)$$ and $$\scriptstyle C_p(\upsilon X)$$, a countably compact subspace of $$\scriptstyle C_p(X)$$ remains countably compact in the topology of $$\scriptstyle C_p(\upsilon X)$$; however, compactness, pseudocompactness, Lindelöf property and Lindelöf $$\scriptstyle \Sigma $$-property can be destroyed by strengthening the topology of $$\scriptstyle C_p(X)$$ to obtain the space $$\scriptstyle C_p(\upsilon X)$$. We show that Lindelöf$$\scriptstyle \Sigma $$-property of $$\scriptstyle C_p(X)$$ together with $$\scriptstyle \omega _1$$ being a caliber of $$\scriptstyle C_p(X)$$ implies that $$\scriptstyle X$$ is cosmic.
Commentationes Mathematicae Universitatis Carolinae, 2009
We study relations between the Lindelof property in the spaces of continuous functions with the t... more We study relations between the Lindelof property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if $C_p(X)$ is Lindelof, $Y=X\cup\{p\}$, and the point $p$ has countable character in $Y$, then $C_p(Y)$ is Lindelof; b) if $Y$ is a cozero subspace of a Tychonoff space $X$, then $l(C_p(Y)^\omega)\le l(C_p(X)^\omega)$ and $\operatorname{ext}(C_p(Y)^\omega)\le \operatorname{ext}(C_p(X)^\omega)$.
Abstract We prove that t m ( X × Y ) ≤ t m ( X ) t m ( Y ) if the space Y is locally compact, and... more Abstract We prove that t m ( X × Y ) ≤ t m ( X ) t m ( Y ) if the space Y is locally compact, and that always t m ( X × Y ) ≤ t m ( X ) χ ( Y ) , where t m ( Z ) is the minitightness (a.k.a. the weak functional tightness) of a space Z .
We prove, modifying Dowker’s example, that there exists a normal space X such that is Lindelof, X... more We prove, modifying Dowker’s example, that there exists a normal space X such that is Lindelof, X is zero-dimensional and is not strongly zero-dimensional.
This chapter provides an overview of tightness and t -equivalence. Two spaces X and Y are called ... more This chapter provides an overview of tightness and t -equivalence. Two spaces X and Y are called M- equivalent if their free topological groups F ( X ) and F ( Y ) in the sense of Markov are topologically isomorphic. The spaces X and Y are l- equivalent if the spaces C p ( X ) and C p ( Y ) of real-valued continuous functions equipped with the topology of point-wise convergence are linearly homeomorphic, and t- equivalent if C p ( X ) and C p ( Y ) are homeomorphic; Arhangel'skiĭ has shown that M -equivalence of two spaces implies their l -equivalence; clearly, l- equivalent spaces are t -equivalent. A topological property is preserved by an equivalence relation if whenever two spaces are in the relation, one of them has the property if and only if the other one does. Similarly, a cardinal invariant is preserved by a relation if its values on two spaces are the same whenever the spaces are in the relation. The chapter discusses about the example that shows the non-preservation of the tightness by M -equivalence depends heavily on the fact that one of the two spaces is not normal. The chapter also discuses problems based on M -equivalent Lindelof spaces.
This chapter provides an overview of LΣ ( κ )-spaces. If κ be a cardinal (finite or infinite), th... more This chapter provides an overview of LΣ ( κ )-spaces. If κ be a cardinal (finite or infinite), then a space X is an LΣ (≤ κ )- space if there is a second-countable space M and a compact-valued upper semi-continuous mapping p : M → X such that p ( M ) = X and w ( p ( x )) ≤ κ for every x ∈ X ( w (·) denotes weight). A space X is an LΣ ( κ )-space if it is an LΣ (≤)‑space and is not an LΣ (≤ λ )-space for any λ κ. A space X is an LΣ ( κ )-space if there is a second-countable space M and a compact-valued upper semi-continuous mapping p : M → X such that p ( M ) = X and w ( p ( x )) κ for every x ∈ X. For finite k the definition says that the images of points under p have at most κ points. The class LΣ (<ω) is the class of all images of second-countable spaces under finite-valued upper semicontinuous mappings. The definitions also admit natural reformulations in terms of networks modulo compact covers in the spirit of the seminal article of K. Nagami.
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