Invariance of domain

Brouwer’s theorem states that every continuous injective map from an open subset of ℝⁿ into ℝⁿ has an open image. It also implies that such a map is a homeomorphism onto its image.

Open map: A function is open if it maps every open set in its domain to an open set in its codomain. Applying the theorem to open subsets shows that every continuous injection between open subsets of ℝⁿ is an open map.
Invariance of dimension: Invariance of dimension states that Euclidean spaces of different finite dimensions are not homeomorphic. It follows by applying invariance of domain to a hypothetical homeomorphism between different-dimensional Euclidean spaces.
L. E. J. Brouwer: L. E. J. Brouwer was a Dutch mathematician who founded intuitionism and made foundational contributions to topology. He proved the theorem now known as invariance of domain.
Space-filling curve: A space-filling curve is a continuous map from an interval onto a region with nonempty interior in the plane. It shows that continuity alone does not prevent a lower-dimensional domain from covering an open region.
Homeomorphism: A homeomorphism is a continuous bijection with a continuous inverse. An injective map in the theorem is a homeomorphism between its domain and its image.
Topological embedding: A topological embedding is a map that is a homeomorphism from its domain onto its image. Every continuous injection covered by the theorem is an embedding, with an open image.
Topology: Topology studies properties of spaces preserved by continuous deformation. Invariance of domain is a landmark result about how continuous maps constrain the shape of spaces.
Continuous bijection: A continuous bijection is a one-to-one and onto function that preserves continuity. Without the theorem’s equal-dimensional open-domain setting, a continuous bijection need not have a continuous inverse.
Local degree: Local degree is an integer measuring the signed number of nearby preimages of a regular value. Degree theory supplies a topological route to proving that injective maps cannot collapse dimension locally.
Invariance of domain in Euclidean spaces: Invariance of domain is the Euclidean theorem that continuous injections between open subsets of equal-dimensional spaces are open. Its open-subset formulation exposes the theorem’s local consequences directly.

Linked from 3 pages