Counting overweight spaces

TL;DR

This paper classifies and constructs a vast diversity of topological spaces with specific properties and sizes, revealing the richness of the space of all such spaces up to homeomorphism.

Contribution

It provides a detailed enumeration and construction of 2^b many topological spaces with various properties, based on infinite cardinal parameters, extending the understanding of space classification.

Findings

Existence of exactly 2^b non-homeomorphic T0-spaces of size a and weight b.

Construction of spaces with diverse properties such as zero-dimensionality, connectedness, and compactness.

Identification of conditions under which certain classes of spaces exist in large quantities.

Abstract

Let c=2^aleph0 denote the cardinality of the continuum and let a,b,k be infinite cardinal numbers with a<b\leq 2^a. We show that there exist precisely 2^b T0-spaces of size a and weight b up to homeomorphism. Among these non-homeomorphic spaces we track down (1) 2^b zero-dimensional, scattered, paracompact, perfectly normal spaces (which are also extremally disconnected in case that b=2^a); (2) 2^b connected and locally connected Hausdorff spaces; (3) 2^b pathwise connected and locally pathwise connected, paracompact, perfectly normal spaces provided that a\geq c; (4) 2^b connected, nowhere locally connected, totally pathwise disconnected, paracompact, perfectly normal spaces provided that a\geq c; (5) 2^b scattered, compact T1-spaces; (6) 2^b connected, locally connected, compact T1-spaces; (7) 2^b pathwise connected and scattered, compact T0-spaces; (8) 2^b scattered, paracompact P_k-spaces whenever a^{<k}=a and b^{<k}=b and 2^b>2^a.