Topology, Metric Spaces and the Generalized Continuum Hypothesis

TL;DR

This paper explores a novel interpretation of set cardinality using Baire Category and topological structures, extending the Baire Category Theorem to spaces with specific properties, and relates this to the Generalized Continuum Hypothesis.

Contribution

It introduces a new topological approach to defining cardinality based on the deletion of clopen sets, connecting it to the GCH and providing models of set theory.

Findings

Cardinality can be characterized by the number of clopen sets needed to delete to empty a set.

Extended Baire Category Theorem applies to $eth$-sequentially complete spaces with no isolated points.

The new cardinality definition aligns with the GCH through specific set-theoretic models.

Abstract

This is a paper that aims to interpret the cardinality of a set in terms of Baire Category, i.e. how many closed nowhere dense sets can be deleted from a set before the set itself becomes negligible. . To do this natural tree-theoretic structures such as the Baire topology are introduced, and the Baire Category Theorem is extended to a statement that a $\aleph$-sequentially complete binary tree representation of a Hausdorff topological space that has a clopen base of cardinality $\aleph$ and no isolated or discrete points is not the union of $<\aleph+1$-many nowhere dense subsets for cardinal $\aleph\ge\aleph_{0}$, where a $\aleph$-sequentially complete topological space is a space where every function $f:\aleph\rightarrow\{0.1\}$ is such that $(\forall x)(x\in f\rightarrow x\in\in X)\rightarrow(f\in X)$. It is shown that if $\aleph<\left|X\right|\le2^{\aleph}$ for $\left|X\right|$ the cardinality of a set $X$, then it is possible to force $\left|X\right|-\aleph\times\left|X\right|\ne\emptyset$ by deleting a dense sequence of $\aleph$ specially selected clopen sets, while if any dense sequence of $\aleph+1$ clopen sets are deleted then $\left|X\right|-(\aleph+1)\times\left|X\right|=\emptyset$. This gives rise to an alternative definition of cardinality as the number of basic clopen sets (intervals in fact) needed to be deleted from a set to force an empty remainder. This alternative definition of cardinality is consistent with and follows from the Generalized Continuum Hypothesis, which is shown by exhibiting two models of set theory, one an outer (modal) model, the other an inner, generalized metric model with an information minimization principle.