What is a Dilatable Space in Mathematics?

In mathematics, a dilatable space is a topological space that can be stretched or expanded in some way without changing its underlying structure. More specifically, a dilatable space is a space that admits a continuous self-similarity transformation, meaning that there exists a mapping from the space to itself that preserves the topology but not necessarily the metric.

In other words, a dilatable space is one that can be deformed or stretched in a way that preserves the topological properties of the space, but not necessarily its metric properties (such as distances and angles). This means that the space can be expanded or contracted in some way without changing its fundamental structure or topology.

Examples of dilatable spaces include the real line, the complex plane, and certain other topological spaces. These spaces are all self-similar, meaning that they have a symmetry that allows them to be stretched or compressed in a way that preserves their underlying structure.