# On weakening tightness to weak tightness

## TL;DR

This paper explores the concept of weak tightness in topological spaces, demonstrating its differences from tightness, and extends several classical theorems to this new setting with new bounds and properties.

## Contribution

It introduces the notion of weak tightness, provides counterexamples to existing theorems, and extends classical results to spaces with weak tightness, refining bounds on cardinal invariants.

## Key findings

- Counterexample showing weak tightness can be strictly less than tightness.
- Refinement of bounds on the size of homogeneous compacta using weak tightness.
- Extension of classical theorems to spaces with weak tightness, including bounds on density and number of G_k-points.

## Abstract

The weak tightness $wt(X)$ of a space $X$ was introduced in [11] with the property $wt(X)\leq t(X)$. We investigate several well-known results concerning $t(X)$ and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum $X$ such that $wt(X)=\aleph_0<t(X)$ under $2^{\aleph_0}=2^{\aleph_1}$. In particular, this demonstrates the celebrated Balogh's Theorem [5] does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if $X$ is a homogeneous compactum then $|X|\leq 2^{wt(X)\pi_\chi(X)}$. This refines a theorem of De la Vega [12]. In the case where the cardinal invariants involved are countable, this also represents a variation of a theorem of Juh\'asz and van Mill [15]. Third, we show that if $X$ is a $T_1$ space, $wt(X)\leq\kappa$, $X$ is $\kappa^+$-compact, and $\psi(\overline{D},X)\leq 2^\kappa$ for any $D\subseteq X$ satisfying $|D|\leq 2^\kappa$, then a) $d(X)\leq 2^\kappa$ and b) $X$ has at most $2^\kappa$-many $G_\kappa$-points. This is a variation of another theorem of Balogh [6]. Finally, we show that if $X$ is a regular space, $\kappa=L(X)wt(X)$, and $\lambda$ is a caliber of $X$ satisfying $\kappa<\lambda\leq \left(2^{\kappa}\right)^+$, then $d(X)\leq 2^{\kappa}$. This extends of theorem of Arhangel'skii [3].