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A note on g -metrizable spaces

Jinjin Li (2003)

Czechoslovak Mathematical Journal

In this paper, the relationships between metric spaces and g -metrizable spaces are established in terms of certain quotient mappings, which is an answer to Alexandroff’s problems.

A note on G δ ideals of compact sets

Maya Saran (2009)

Commentationes Mathematicae Universitatis Carolinae

Solecki has shown that a broad natural class of G δ ideals of compact sets can be represented through the ideal of nowhere dense subsets of a closed subset of the hyperspace of compact sets. In this note we show that the closed subset in this representation can be taken to be closed upwards.

A note on generic chaos

Gongfu Liao (1994)

Annales Polonici Mathematici

We consider dynamical systems on a separable metric space containing at least two points. It is proved that weak topological mixing implies generic chaos, but the converse is false. As an application, some results of Piórek are simply reproved.

A note on inverse limits of continuous images of arcs.

Ivan Loncar (1999)

Publicacions Matemàtiques

The main purpose of this paper is to prove some theorems concerning inverse systems and limits of continuous images of arcs. In particular, we shall prove that if X = {Xa, pab, A} is an inverse system of continuous images of arcs with monotone bonding mappings such that cf (card (A)) ≠ w1, then X = lim X is a continuous image of an arc if and only if each proper subsystem {Xa, pab, B} of X with cf(card (B)) = w1 has the limit which is a continuous image of an arc (Theorem 18).

A note on k-c-semistratifiable spaces and strong β -spaces

Li-Xia Wang, Liang-Xue Peng (2011)

Mathematica Bohemica

Recall that a space X is a c-semistratifiable (CSS) space, if the compact sets of X are G δ -sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a T 2 -space X is a k-c-semistratifiable space if and only if X has a g function which satisfies the following conditions: (1) For each x ∈ X , { x } = ⋂ { g ( x , n ) : n ∈ ℕ } and g ( x , n + 1 ) ⊆ g ( x , n ) for each n ∈ ℕ . (2) If a...

A note on linear mappings between function spaces

Jan Baars (1993)

Commentationes Mathematicae Universitatis Carolinae

Arhangel’skiǐ proved that if X and Y are completely regular spaces such that C p ( X ) and C p ( Y ) are linearly homeomorphic, then X is pseudocompact if and only if Y is pseudocompact. In addition he proved the same result for compactness, σ -compactness and realcompactness. In this paper we prove that if φ : C p ( X ) → C p ( X ) is a continuous linear surjection, then Y is pseudocompact provided X is and if φ is a continuous linear injection, then X is pseudocompact provided Y is. We also give examples that both statements do not hold...

A note on monotone countable paracompactness

Ge Ying, Chris Good (2001)

Commentationes Mathematicae Universitatis Carolinae

We show that a space is MCP (monotone countable paracompact) if and only if it has property ( * ) , introduced by Teng, Xia and Lin. The relationship between MCP and stratifiability is highlighted by a similar characterization of stratifiability. Using this result, we prove that MCP is preserved by both countably biquotient closed and peripherally countably compact closed mappings, from which it follows that both strongly Fréchet spaces and q-space closed images of MCP spaces are MCP. Some results on...

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