The complement of a $\sigma$-compact subset of a space with a $\pi$-tree also has a $\pi$-tree

TL;DR

This paper proves that the complement of a $\sigma$-compact subset in a space with a $\pi$-tree also has a $\pi$-tree, introducing a new foliage hybrid operation to modify trees accordingly.

Contribution

It introduces the foliage hybrid operation and demonstrates how to construct a $\pi$-tree for subspaces by modifying existing trees.

Findings

The complement of a $\sigma$-compact subset retains a $\pi$-tree.

The foliage hybrid operation effectively modifies trees for subspace construction.

The method applies to a broad class of topological spaces with $\pi$-trees.

Abstract

We prove that the complement of a $\sigma$-compact subset of a topological space that has a $\pi$-tree also has a $\pi$-tree. To do this, we construct the foliage hybrid operation, which deals with foliage trees (that is, set-theoretic trees with a `leaf' at each node). Then using this operation we modify a $\pi$-tree of a space and get a $\pi$-tree for its subspace.