# Properness under closed forcing

## TL;DR

This paper explores how certain proper posets can become improper after applying specific $ ext{κ}$-closed forcing extensions, revealing nuanced behavior of properness under forcing.

## Contribution

It provides explicit examples of proper posets that lose their properness in some $ ext{κ}$-closed forcing extensions, highlighting limitations of properness preservation.

## Key findings

- Proper posets can become improper after $ ext{κ}$-closed forcing.
- Examples demonstrate properness is not always preserved under certain extensions.
- Properness behavior varies with different forcing extensions.

## Abstract

For every uncountable regular $\kappa$, we give two examples of proper posets which turn improper in some $\kappa$-closed forcing extension.