local dimension of a locally Euclidean space

Let X be a locally Euclidean space. Recall that the local dimensionMathworldPlanetmathPlanetmath of X in yX is a natural numberMathworldPlanetmath n such that there is an open neighbourhood UX of y homeomorphic to n. This number is well defined (please, see parent object for more details) and we will denote it by dimyX.

PropositionPlanetmathPlanetmathPlanetmath. Function f:X defined by f(y)=dimyX is continuousPlanetmathPlanetmath (where on we have discrete topology).

Proof. It is enough to show that preimageMathworldPlanetmath of a point is open. Assume that n and yX is such that f(y)=n. Then there is an open neighbourhood UX of y such that U is homeomorphic to n. Obviously for any xU we have that U is an open neighbourhood of x homeomorphic to n. Therefore f(x)=n, so Uf-1(n). Thus (since y was arbitrary) we’ve shown that around every point in f-1(n) there is an open neighbourhood of that point contained in f-1(n). This shows that f-1(n) is open, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Corollary. Assume that X is a connectedPlanetmathPlanetmath, locally Euclidean space. Then local dimension is constant, i.e. there exists natural number n such that for any yX we have


Proof. Consider the mapping f:X such that f(y)=dimyX. Proposition shows that f is continuous. Therefore f(X) is connected, because X is. But has discrete topology, so there are no other connected subsets then points. Thus there is n such that f(X)={n}, which completes the proof.

Remark. Generally, local dimension need not be constant. For example consider X1,X23 such that

X1={(x,0,0)|x}  X2={(x,y,1)|x,y}.

One can easily show that X=X1X2 (with topologyMathworldPlanetmathPlanetmath inherited from 3) is locally Euclidean, but dim(0,0,0)X=1 and dim(1,1,1)X=2.

Title local dimension of a locally Euclidean space
Canonical name LocalDimensionOfALocallyEuclideanSpace
Date of creation 2013-03-22 18:55:34
Last modified on 2013-03-22 18:55:34
Owner joking (16130)
Last modified by joking (16130)
Numerical id 7
Author joking (16130)
Entry type Theorem
Classification msc 53-00