# On the preservation of Baire and weakly Baire category

Alireza Kamel Mirmostafaee; Zbigniew Piotrowski

Mathematica Bohemica (2016)

- Volume: 141, Issue: 4, page 475-481
- ISSN: 0862-7959

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topMirmostafaee, Alireza Kamel, and Piotrowski, Zbigniew. "On the preservation of Baire and weakly Baire category." Mathematica Bohemica 141.4 (2016): 475-481. <http://eudml.org/doc/287586>.

@article{Mirmostafaee2016,

abstract = {We consider the question of preservation of Baire and weakly Baire category under images and preimages of certain kind of functions. It is known that Baire category is preserved under image of quasi-continuous feebly open surjections. In order to extend this result, we introduce a strictly larger class of quasi-continuous functions, i.e. the class of quasi-interior continuous functions. We show that Baire and weakly Baire categories are preserved under image of feebly open quasi-interior continuous surjections. We also give a new definition for countably fiber-completeness of a function. We prove that Baire category is preserved under inverse image of a countably fiber-complete function provided that it is feebly open and feebly continuous.},

author = {Mirmostafaee, Alireza Kamel, Piotrowski, Zbigniew},

journal = {Mathematica Bohemica},

keywords = {feebly continuous mapping; quasi-interior continuity; Baire space; weakly Baire space; fiber-completeness},

language = {eng},

number = {4},

pages = {475-481},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {On the preservation of Baire and weakly Baire category},

url = {http://eudml.org/doc/287586},

volume = {141},

year = {2016},

}

TY - JOUR

AU - Mirmostafaee, Alireza Kamel

AU - Piotrowski, Zbigniew

TI - On the preservation of Baire and weakly Baire category

JO - Mathematica Bohemica

PY - 2016

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 141

IS - 4

SP - 475

EP - 481

AB - We consider the question of preservation of Baire and weakly Baire category under images and preimages of certain kind of functions. It is known that Baire category is preserved under image of quasi-continuous feebly open surjections. In order to extend this result, we introduce a strictly larger class of quasi-continuous functions, i.e. the class of quasi-interior continuous functions. We show that Baire and weakly Baire categories are preserved under image of feebly open quasi-interior continuous surjections. We also give a new definition for countably fiber-completeness of a function. We prove that Baire category is preserved under inverse image of a countably fiber-complete function provided that it is feebly open and feebly continuous.

LA - eng

KW - feebly continuous mapping; quasi-interior continuity; Baire space; weakly Baire space; fiber-completeness

UR - http://eudml.org/doc/287586

ER -

## References

top- Beer, G., Villar, L., Weakly Baire spaces, Southeast Asian Bull. Math. 11 (1988), 127-133. (1988) Zbl0665.54019MR0958315
- Bourbaki, N., Topologie Générale -- Chapitre 9: Utilisation des Nombres Réels en Topologie Générale, Éléments de Mathématique I: Les Structures Fondamentales de L'analyse -- Livre III Actualités Scientifiques et Industrielles, No. 1045 Hermann & Cie, Paris French (1948). (1948) MR0027138
- Cao, J., Moors, W. B., A survey on topological games and their applications in analysis, RACSAM, Rev. R. Acad. Cienc. Exactas Fí s. Nat. Ser. A Mat. 100 (2006), 39-49. (2006) Zbl1114.91024MR2267399
- Choquet, G., Lectures on Analysis, vol. 1: Integration and Topological Vector Spaces, Mathematics Lecture Note Series W. A. Benjamin, New York-Amsterdam (1969). (1969) Zbl0181.39601MR0250011
- Dobo{š}, J., A note on the invariance of Baire spaces under mappings, Časopis Pěst. Mat. 108 (1983), 409-411. (1983) Zbl0535.54005MR0727538
- Doboš, J., Piotrowski, Z., Reilly, I. L., Preimages of Baire spaces, Math. Bohem. 119 (1994), 373-379. (1994) Zbl0815.54010MR1316589
- Fleissner, W. G., Kunen, K., Barely Baire spaces, Fundam. Math. 101 (1978), 229-240. (1978) Zbl0413.54036MR0521125
- Frol{í}k, Z., Baire spaces and some generalizations of complete metric spaces, Czech. Math. J. 11 (1961), 237-248. (1961) Zbl0149.40302MR0124870
- Frol{í}k, Z., Remarks concerning the invariance of Baire spaces under mappings, Czech. Math. J. 11 (1961), 381-385. (1961) Zbl0104.17204MR0133098
- Mirmostafaee, A. K., 10.2478/s12175-014-0255-1, Math. Slovaca 64 (2014), 1019-1026. (2014) Zbl1349.54034MR3255869DOI10.2478/s12175-014-0255-1
- Moors, W. B., 10.1090/S0002-9939-06-08389-4, Proc. Am. Math. Soc. 134 (2006), 2161-2163. (2006) Zbl1093.54008MR2215788DOI10.1090/S0002-9939-06-08389-4
- Neubrunn, T., A note on mappings of Baire spaces, Math. Slovaca 27 (1977), 173-176 correction in 442 (1977). (1977) Zbl0371.54023MR0454910
- Noll, D., 10.1090/S0002-9939-1989-0982407-2, Proc. Am. Math. Soc. 107 (1989), 847-854. (1989) Zbl0687.54012MR0982407DOI10.1090/S0002-9939-1989-0982407-2
- Oxtoby, J. C., Cartesian products of Baire spaces, Fundam. Math. 49 (1961), 157-166. (1961) Zbl0113.16402MR0140638
- Oxtoby, J. C., Measure and Category---A Survey of the Analogies between Topological and Measure Spaces, Graduate Texts in Mathematics. Vol. 2 Springer, New York (1971). (1971) Zbl0217.09201MR0393403
- Piotrowski, Z., Reilly, I. L., Preimages of Baire spaces---an example, Quest. Answers Gen. Topology 11 (1993), 105-107. (1993) Zbl0779.54025MR1205956
- Rose, D. A., Jankovi{ć}, D. S., Hamlett, T. R., On weakly Baire spaces, Southeast Asian Bull. Math. 15 (1991), 183-190. (1991) Zbl0747.54008MR1145440
- Rudin, W., Functional Analysis, International Series in Pure and Applied Mathematics McGraw-Hill, New York (1991). (1991) Zbl0867.46001MR1157815
- Sikorski, R., On the Cartesian product of metric spaces, Fundam. Math. 34 (1947), 288-292. (1947) Zbl0041.31701MR0025542

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