# Continuous extension of maps between sequential cascades

## TL;DR

This paper investigates the properties of contours in sequential cascades, explores contour inversion, and provides a general solution for extending maps between maximal elements to subcascades.

## Contribution

It introduces a comprehensive method for continuous extension of maps between maximal elements of sequential cascades to their subcascades.

## Key findings

- Contour inversion for iterated contours is characterized.
- A general solution for extending maps between maximal elements is provided.
- Topologicity and regularity of convergences are analyzed using contour operations.

## Abstract

The contour of a family of filters along a filter is a set-theoretic lower limit. Topologicity and regularity of convergences can be characterized with the aid of the contour operation. Contour inversion is studied, in particular, for iterated contours of sequential cascades. A related problem of continuous extension of maps between maximal elements of sequential cascades to full subcascades is solved in full generality.