In this paper, countable compactness and the Lindel¨of propertyare defined for L-fuzzy sets, where L is a complete de Morgan algebra. Theydon’t rely on the structure of the basis lattice L and no distributivity is requiredin L. A fuzzy compact L-set is countably compact and has the Lindel¨ofproperty. An L-set having the Lindel¨of property is countably compact if andonly if it is fuzzy compact. ...

In this paper, we prove the following statements: (1) There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable. (2) Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace. (3) Assuming 20 = 21 , there exists a normal absolutely star countable discrete clos...

It was proved in McKinsey and Tarski [7] that every finite wellconnected closure algebra is embedded into the closure algebra of the power set of the real line R. Pucket [10] extended this result to all finite connected closure algebras by showing that there exists an open map fromR to any finite connected topological space. We simplify his proof considerably by using the correspondence between...

By a classical theorem of Harvey Friedman (1973), every countable nonstandard model $\mathcal{M}$ sufficiently strong fragment ZF has proper rank-initial self-embedding $j$, i.e., $j$ is such that $j[\mathcal{M}]\subsetneq\mathcal{M}$, and the ordinal rank each member $j[\mathcal{M}]$ less than element $\mathcal{M}\setminus j[\mathcal{M}]$. Here we investigate larger family initial-embeddings m...

In this paper, we discuss the relationships between spaces with locally countable weak-bases and spaces with various locally countable networks, establish the relationships between spaces with locally countable weak-bases and locally separable metric spaces, and show that spaces have locally countable weak-bases if and only if they are locally Lindelöf, g-metrizable spaces. These are improvemen...

We present a countable complete first order theory T which is model theoretically very well behaved: it eliminates quantifiers, is ω-stable, it has NDOP and is shallow of depth two. On the other hand, there is no countable bound on the Scott heights of its countable models, which implies that the isomorphism relation for countable models is not Borel.