COUNTING SPACES OF EXCESSIVE WEIGHTS

Abstract

Let $\kappa,\lambda$ be infinite cardinal numberswith $\kappa<\lambda\leq 2^\kappa$.We show that there exist precisely $2^\lambda$ T$_0$-spacesof size $\kappa$ and weight $\lambda$up to homeomorphism. Among these non-homeomorphic spaces we track down(i) $2^{\lambda}$ zero-dimensional,scattered, para\-compact, perfectly normal spaces(which are also extremally disconnected in casethat $\lambda=2^\kappa$);(ii) $2^{\lambda}$connected and locally connected Hausdorff spaces;(iii) $2^{\lambda}$ pathwise connected and locally pathwise connected,paracompact, perfectly normal spacesprovided that $\kappa\geq 2^{\aleph_0}$;(iv) $2^{\lambda}$ connected,nowhere locally connected, totally pathwise disconnected,paracompact, perfectly normal spacesprovided that $\kappa\geq 2^{\aleph_0}$;(v) $2^\lambda$ scattered, compact T$_1$-spaces;(vi) $2^\lambda$ connected, locally connected, compact T$_1$-spaces;(vii) $2^\lambda$ pathwise connected {\it and} scattered,compact T$_0$-spaces;(viii) $2^\lambda$ scattered, paracompact$P_\alpha$-spaces whenever $\kappa^{<\alpha}=\kappa$and $\lambda^{<\alpha}=\lambda$ and $2^\lambda>2^\kappa$.