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1

Between local connectedness and sum connectedness

Kohli J. K., Singh D., Tyagi B. K.

Commentationes Mathematicae

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2013

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Vol 53, No. 1

3--16

EN

A new generalization of local connectedness called Z-local connectedness is introduced. Basic properties of Z-locally connected spaces are studied and their place in the hierarchy of variants of local connectedness, which already exist in the literature, is elaborated. The class of Z-locally connected spaces lies strictly between the classes of pseudo locally connected spaces (Commentations Math. 50(2)(2010),183-199) and sum connected spaces ( weakly locally connected spaces) (Math. Nachrichten 82(1978), 121-129; Ann. Acad. Sci. Fenn. AI Math. 3(1977), 185- 205) and so contains all quasi locally connected spaces which in their turn contain all almost locally connected spaces introduced by Mancuso (J. Austral. Math. Soc. 31(1981), 421-428). Formulations of product and subspace theorems for Z-locally connected spaces are suggested. Their preservation under mappings and their interplay with mappings are discussed. Change of topology of a Z-locally connected space is considered so that it is simply a locally connected space in the coarser topology. It turns out that the full subcategory of Z-locally connected spaces provides another example of a mono-coreflective subcategory of TOP which properly contains all almost locally connected spaces.

2

On BPI Restricted to Boolean Algebras of Size Continuum

Hall E., Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2013

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Vol. 61, no 1

9--21

EN

(i) The statement P(ω)=“every partition of R has size ≤|R|” is equivalent to the proposition R(ω)=“for every subspace Y of the Tychonoff product 2P(ω) the restriction B|Y={Y∩B:B∈B} of the standard clopen base B of 2P(ω) to Y has size ≤|P(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of P(ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤|R| every filter can be extended to an ultrafilter. (b) Every Boolean algebra of size ≤|R| has an ultrafilter.

3

(ĝ,s)-continuous functions between topological spaces

Ravi O, Thivagar M. L., Latha R

Fasciculi Mathematici

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2011

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Nr 46

115--130

EN

In this paper, we introduce (ĝ,s)-continuous functions between topological spaces, study some of its basic properties and discuss its relationships with other topological functions.

4

On some classes of sets in extremally disconnected spaces

Duszyński Z.

Demonstratio Mathematica

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2011

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Vol. 44, nr 1

169--180

EN

In the present paper several characterizations of the classical notion of extremally disconnected spaces are obtained. A few relationships for finite products of extremally disconnected spaces are also studied.

5

Quasi Locally Connected Spaces and Pseudo Locally Connected Spaces

Kohli J. K., Singh D., Tyagi B. K.

Commentationes Mathematicae

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2010

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Vol. 50, [Z] 2

183-199

EN

Two new generalizations of locally connected spaces called "quasi locally connected spaces" and "pseudo locally connected spaces" are introduced and their basic properties are studied. The class of quasi locally connected spaces properly contains the class of almost locally connected spaces (J. Austral. Math. Soc. 31(1981), 421–428) and is strictly contained in the class of pseudo locally connected spaces which in its turn is properly contained in the class of sum connected spaces (Math.Nachrichten 82(1978), 121-129; Ann. Acad. Sci. Fenn. A I Math. 3(1977), 185–205). Product and subspace theorems for quasi (pseudo) locally connected spaces are discussed. Their preservation under mappings and their interplay with mappings are outlined. Function spaces of quasi (pseudo) locally connected spaces are considered. Change of topology of a quasi (pseudo) locally connected space is considered so that it is simply a locally connected space in the coarser topology. In contradistinction with almost locally connected spaces, quasi (pseudo) locally connected spaces constitute a coreflective subcategory of TOP.