COUNTING SPACES OF EXCESSIVE WEIGHTS

Abstract

Let $\kappa ,\lambda$ be infinite cardinal numberswith $\kappa <\lambda \le 2^{\kappa }$.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 $\kappa \ge 2^{{\mathrm{\aleph }}_0}$;(iv) $2^{\lambda }$ connected,nowhere locally connected, totally pathwise disconnected,paracompact, perfectly normal spacesprovided that $\kappa \ge 2^{{\mathrm{\aleph }}_0}$;(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 $2^{\lambda }>2^{\kappa }$.