# A common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequality

## TL;DR

This paper introduces a unified bound for the weak Lindelöf number of the $G_\delta$-modification of Hausdorff spaces, generalizing key cardinal inequalities in topology without requiring separation axioms beyond $T_2$.

## Contribution

It provides a common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequality, answering a longstanding open question in cardinal invariants.

## Key findings

- Unified bound for the weak Lindelöf number of $G_\delta$-modification
- Generalizes Arhangel'skii's and Hajnal-Juhasz inequalities
- Does not require separation axioms beyond $T_2$

## Abstract

We present a bound for the weak Lindel\"of number of the $G_\delta$-modification of a Hausdorff space which implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\le 2^{L(X)\chi(X)}$ (Arhangel'skii) and $|X|\le 2^{c(X)\chi (X)}$ (Hajnal-Juhasz). This solves a question that goes back to Bell, Ginsburg and Woods and is mentioned in Hodel's survey on Arhangel'skii's Theorem. In contrast to previous attempts we do not need any separation axiom beyond $T_2$.