Abstract

Let $\kappa,\lambda$ be infinite cardinal numbers with $\kappa<\lambda\leq 2^\kappa$. We show that there exist precisely $2^\lambda$ T$_0$-spaces of 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 case that $\lambda=2^\kappa$); (ii) $2^{\lambda}$ connected and locally connected Hausdorff spaces; (iii) $2^{\lambda}$ pathwise connected and locally pathwise connected, paracompact, perfectly normal spaces provided that $\kappa\geq 2^{\aleph_0}$; (iv) $2^{\lambda}$ connected, nowhere locally connected, totally pathwise disconnected, paracompact, perfectly normal spaces provided 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 $2^\lambda>2^\kappa$.

Keywords: Scattered resp. connected; paracompact spaces.

MSC: 54A525, 54D20, 54D05

Pages:  165--186     

Volume  72 ,  Issue  2 ,  2020