I'm willing to take that compositions of continuous functions are continuous without proof. I know that sums and products are continuous, but only when the domain is some subset of R^n. Does this carry over to any domain? If that's the case, then there is nothing to show.mathwonk said:what do you know about continuous functions? compositions of them are still continuous, as are sums, products,...
A topological space is a mathematical concept that allows us to study the properties of spaces that are defined by their structure, rather than their specific geometric properties.
A metric space is a type of topological space, where a distance function (or metric) is defined between points. In a topological space, the notion of distance may not be defined, but the properties of connectedness, continuity, and convergence can still be studied.
Some common examples of topological spaces include the real line, Euclidean spaces, and the surface of a sphere. Other examples include graph structures, networks, and manifolds.
Topological spaces have applications in various fields such as physics, biology, and computer science. They provide a powerful framework for analyzing complex systems and understanding their behavior and properties.
Topological spaces are studied using various techniques such as algebraic topology, differential topology, and geometric topology. Visualization techniques such as diagrams, graphs, and 3D models can also be used to understand the structure and properties of topological spaces.