Continuous bijection from a compact space to a Hausdorff space is a homeomorphism
Builds on Compact subsets of a Hausdorff space are closedClosed subsets of compact sets are compactContinuous image of a compact set is compactHomeomorphismContinuity via closed sets
Statement
Let be a compact space, let be a Hausdorff space and let be a continuous map that is bijective. Then is a homeomorphism.
Remarks
Proof: it is enough to show that maps closed sets to closed sets. A closed is compact because closed subsets of compact sets are compact. Then is compact, and thus closed in by compact subsets of a Hausdorff space are closed. The compactness is needed: the map from to the unit circle is a continuous bijection, but its inverse is not continuous.
Reading path
To understand this page from the ground up, read these 49 pages in order. The foundations come first.
- Element definition
- Proposition concept
- Real numbers object
- Set definition
- Conjunction definition
- Disjunction definition
- Implication definition
- Predicate definition
- Empty set object
- Family of sets definition
- Ordered pair definition
- Intersection definition
- Union definition
- Existential quantifier definition
- Universal quantifier definition
- Cartesian product definition
- Subset definition
- Union of a family of sets definition
- Uniqueness quantifier definition
- Binary relation definition
- Universal set definition
- Topological space definition
- Complement definition
- Function definition
- Subspace topology definition
- Codomain definition
- Injective function definition
- Metric space definition
- Preimage of a set definition
- Surjective function definition
- Open ball definition
- Bijective function definition
- Metric topology definition
- Inverse function definition
- Open set definition
- Closed set definition
- Hausdorff space definition
- Open cover definition
- Continuous map of topological spaces definition
- Complement of an open set is closed statement
- Homeomorphism definition
- Compact space definition
- Continuity via closed sets statement
- Compact set definition
- Compactness via open covers statement
- Closed subsets of compact sets are compact statement
- Compact subsets of a Hausdorff space are closed statement
- Continuous image of a compact set is compact statement
- Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
Where this leads
21 pages build on this one.
- Torus
- Cantor set is homeomorphic to the space of binary sequences
- Circle as a quotient of the real line
- Translation surface
- Classification of surfaces
- Brouwer's characterization of the Cantor space
- Kolmogorov–Arnold–Moser theorem
- Standard map
- Arnold's cat map
- Action-angle coordinates
- Sinai billiard
- Milnor's isospectral flat tori
- Torus knot
- Liouville–Arnold theorem
- Cantor algebra
- Cantor space
- Pseudo-Anosov map
- Algorithmic randomness
- Torus as a group
- Bunimovich stadium
- Maximal torus
Further reading
- Web https://en.wikipedia.org/wiki/Homeomorphism, Properties
About this page
| Tag | 0430ETCMKD |
|---|---|
| URI | m2://0430ETCMKD |
| Kind | statement, theorem |
| Writing | W0 stub |
| MSC | 54D30 |
| Status | This copy of the site has no build data. Run m2 status 0430ETCMKD. |
| Uses | Compact subsets of a Hausdorff space are closed, Closed subsets of compact sets are compact, Continuous image of a compact set is compact, Homeomorphism, Continuity via closed sets |
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