A continuous injective map from an open subset of n-dimensional Euclidean space into n-dimensional Euclidean space is itself an open map, and hence a homeomorphism onto its image. Proved by L. E. J. Brouwer, it shows that dimension and openness are topologically robust notions.
Facts
StatementIf U is an open subset of n-dimensional Euclidean space and f is an injective continuous map from U into that same space, then the image of U under f is open and f is a homeomorphism between U and its image. 1 Connections
Sources
1. Invariance of Domain (Wikipedia)
Wikimedia Foundationlead paragraph, topic sentence
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space.
lead paragraph, attribution and publication sentence
The theorem and its proof are due to L. E. J. Brouwer, published in 1912.
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