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An independency result in connectification theory

Alessandro Fedeli, Attilio Le Donne (1999)

Commentationes Mathematicae Universitatis Carolinae

A space is called connectifiable if it can be densely embedded in a connected Hausdorff space. Let ψ be the following statement: “a perfect T 3 -space X with no more than 2 𝔠 clopen subsets is connectifiable if and only if no proper nonempty clopen subset of X is feebly compact". In this note we show that neither ψ nor ¬ ψ is provable in ZFC.

Diagonals and discrete subsets of squares

Dennis Burke, Vladimir Vladimirovich Tkachuk (2013)

Commentationes Mathematicae Universitatis Carolinae

In 2008 Juhász and Szentmiklóssy established that for every compact space X there exists a discrete D X × X with | D | = d ( X ) . We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf Σ -space X and hence X ω is d -separable. We give an example of a countably compact space X such that X ω is not d -separable. On the other hand, we show that for any Lindelöf p -space X there exists a discrete subset D X × X such that Δ = { ( x , x ) : x X } D ¯ ; in particular, the diagonal Δ is a retract of D ¯ and the projection...

Further remarks on KC and related spaces

Angelo Bella, Camillo Costantini (2011)

Commentationes Mathematicae Universitatis Carolinae

A topological space is KC when every compact set is closed and SC when every convergent sequence together with its limit is closed. We present a complete description of KC-closed, SC-closed and SC minimal spaces. We also discuss the behaviour of the finite derived set property in these classes.

H-closed functions

Filippo Cammaroto, Vitaly V. Fedorcuk, Jack R. Porter (1998)

Commentationes Mathematicae Universitatis Carolinae

The notion of a Hausdorff function is generalized to the concept of H-closed function and the concept of an H-closed extension of a Hausdorff function is developed. Each Hausdorff function is shown to have an H-closed extension.

N -sets and near compact spaces

Filippo Cammaroto, Giovanni Lo Faro, Jack R. Porter (1999)

Bollettino dell'Unione Matematica Italiana

Si provano nuovi risultati riguardanti gli « N -sets» e gli spazi «Near-compact». Si completano alcune ricerche pubblicate dai primi due autori nel 1978 e si risolvono due problemi recentemente posti da Cammaroto, Gutierrez, Nordo e Prada.

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