## Strong $$\beta$$-spaces and their countable products.(English)Zbl 1243.54046

Summary: We introduce a new concept called “strong beta-spaces”. Strong beta-spaces are beta-spaces. Semi-stratifiable spaces, strong Sigma-spaces and strict $$p$$-spaces are all strong beta-space. Thus, the class of strong beta-spaces is well located in the classes of generalized metric spaces. This class does not coincide with the class of beta-spaces, but they coincide under the condition of paracompactness. As a merit of the class of strong beta-spaces, we show that it is countably productive. Moreover, it is shown that the class of normal strong beta-spaces (or paracompact beta-spaces) is countably productive if it is finitely productive.

### MSC:

 54E20 Stratifiable spaces, cosmic spaces, etc. 54B10 Product spaces in general topology 54E18 $$p$$-spaces, $$M$$-spaces, $$\sigma$$-spaces, etc.