A topological manifold is a mathematical space that resembles Euclidean space near each point. More formally, it is a topological space that is locally homeomorphic to ℝ^n, meaning that around every point, there exists a neighborhood that can be mapped to an open set in ℝ^n. This property allows for the study of shapes and spaces in a flexible way, without requiring a strict geometric structure. Topological manifolds can be classified by their dimensions, such as 1-dimensional manifolds like circles or 2-dimensional manifolds like surfaces. They are fundamental in various fields, including geometry, physics, and algebraic topology, as they provide a framework for understanding complex shapes and their properties.