Local homeomorphism

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In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If is a local homeomorphism, is said to be an étale space over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.

A topological space is locally homeomorphic to if every point of has a neighborhood that is homeomorphic to an open subset of For example, a manifold of dimension is locally homeomorphic to

If there is a local homeomorphism from to then is locally homeomorphic to but the converse is not always true. For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane but there is no local homeomorphism

Formal definition

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A function   between two topological spaces is called a local homeomorphism[1] if every point   has an open neighborhood   whose image   is open in   and the restriction   is a homeomorphism (where the respective subspace topologies are used on   and on  ).

Examples and sufficient conditions

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Local homeomorphisms versus homeomorphisms

Every homeomorphism is a local homeomorphism. But a local homeomorphism is a homeomorphism if and only if it is bijective. A local homeomorphism need not be a homeomorphism. For example, the function   defined by   (so that geometrically, this map wraps the real line around the circle) is a local homeomorphism but not a homeomorphism. The map   defined by   which wraps the circle around itself   times (that is, has winding number  ), is a local homeomorphism for all non-zero   but it is a homeomorphism only when it is bijective (that is, only when   or  ).

Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover   of a space   is a local homeomorphism. In certain situations the converse is true. For example: if   is a proper local homeomorphism between two Hausdorff spaces and if   is also locally compact, then   is a covering map.

Local homeomorphisms and composition of functions

The composition of two local homeomorphisms is a local homeomorphism; explicitly, if   and   are local homeomorphisms then the composition   is also a local homeomorphism. The restriction of a local homeomorphism to any open subset of the domain will again be a local homomorphism; explicitly, if   is a local homeomorphism then its restriction   to any   open subset of   is also a local homeomorphism.

If   is continuous while both   and   are local homeomorphisms, then   is also a local homeomorphism.

Inclusion maps

If   is any subspace (where as usual,   is equipped with the subspace topology induced by  ) then the inclusion map   is always a topological embedding. But it is a local homeomorphism if and only if   is open in   The subset   being open in   is essential for the inclusion map to be a local homeomorphism because the inclusion map of a non-open subset of   never yields a local homeomorphism (since it will not be an open map).

The restriction   of a function   to a subset   is equal to its composition with the inclusion map   explicitly,   Since the composition of two local homeomorphisms is a local homeomorphism, if   and   are local homomorphisms then so is   Thus restrictions of local homeomorphisms to open subsets are local homeomorphisms.

Invariance of domain

Invariance of domain guarantees that if   is a continuous injective map from an open subset   of   then   is open in   and   is a homeomorphism. Consequently, a continuous map   from an open subset   will be a local homeomorphism if and only if it is a locally injective map (meaning that every point in   has a neighborhood   such that the restriction of   to   is injective).

Local homeomorphisms in analysis

It is shown in complex analysis that a complex analytic function   (where   is an open subset of the complex plane  ) is a local homeomorphism precisely when the derivative   is non-zero for all   The function   on an open disk around   is not a local homeomorphism at   when   In that case   is a point of "ramification" (intuitively,   sheets come together there).

Using the inverse function theorem one can show that a continuously differentiable function   (where   is an open subset of  ) is a local homeomorphism if the derivative   is an invertible linear map (invertible square matrix) for every   (The converse is false, as shown by the local homeomorphism   with  ). An analogous condition can be formulated for maps between differentiable manifolds.

Local homeomorphisms and fibers

Suppose   is a continuous open surjection between two Hausdorff second-countable spaces where   is a Baire space and   is a normal space. If every fiber of   is a discrete subspace of   (which is a necessary condition for   to be a local homeomorphism) then   is a  -valued local homeomorphism on a dense open subset of   To clarify this statement's conclusion, let   be the (unique) largest open subset of   such that   is a local homeomorphism.[note 1] If every fiber of   is a discrete subspace of   then this open set   is necessarily a dense subset of   In particular, if   then   a conclusion that may be false without the assumption that  's fibers are discrete (see this footnote[note 2] for an example). One corollary is that every continuous open surjection   between completely metrizable second-countable spaces that has discrete fibers is "almost everywhere" a local homeomorphism (in the topological sense that   is a dense open subset of its domain). For example, the map   defined by the polynomial   is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset   is dense in   with additional effort (using the inverse function theorem for instance), it can be shown that   which confirms that this set is indeed dense in   This example also shows that it is possible for   to be a proper dense subset of  's domain. Because every fiber of every non-constant polynomial is finite (and thus a discrete, and even compact, subspace), this example generalizes to such polynomials whenever the mapping induced by it is an open map.[note 3]

Local homeomorphisms and Hausdorffness

There exist local homeomorphisms   where   is a Hausdorff space but   is not. Consider for instance the quotient space   where the equivalence relation   on the disjoint union of two copies of the reals identifies every negative real of the first copy with the corresponding negative real of the second copy. The two copies of   are not identified and they do not have any disjoint neighborhoods, so   is not Hausdorff. One readily checks that the natural map   is a local homeomorphism. The fiber   has two elements if   and one element if   Similarly, it is possible to construct a local homeomorphisms   where   is Hausdorff and   is not: pick the natural map from   to   with the same equivalence relation   as above.

Properties

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A map is a local homeomorphism if and only if it is continuous, open, and locally injective. In particular, every local homeomorphism is a continuous and open map. A bijective local homeomorphism is therefore a homeomorphism.

Whether or not a function   is a local homeomorphism depends on its codomain. The image   of a local homeomorphism   is necessarily an open subset of its codomain   and   will also be a local homeomorphism (that is,   will continue to be a local homeomorphism when it is considered as the surjective map   onto its image, where   has the subspace topology inherited from  ). However, in general it is possible for   to be a local homeomorphism but   to not be a local homeomorphism (as is the case with the map   defined by   for example). A map   is a local homomorphism if and only if   is a local homeomorphism and   is an open subset of  

Every fiber of a local homeomorphism   is a discrete subspace of its domain  

A local homeomorphism   transfers "local" topological properties in both directions:

  •   is locally connected if and only if   is;
  •   is locally path-connected if and only if   is;
  •   is locally compact if and only if   is;
  •   is first-countable if and only if   is.

As pointed out above, the Hausdorff property is not local in this sense and need not be preserved by local homeomorphisms.

The local homeomorphisms with codomain   stand in a natural one-to-one correspondence with the sheaves of sets on   this correspondence is in fact an equivalence of categories. Furthermore, every continuous map with codomain   gives rise to a uniquely defined local homeomorphism with codomain   in a natural way. All of this is explained in detail in the article on sheaves.

Generalizations and analogous concepts

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The idea of a local homeomorphism can be formulated in geometric settings different from that of topological spaces. For differentiable manifolds, we obtain the local diffeomorphisms; for schemes, we have the formally étale morphisms and the étale morphisms; and for toposes, we get the étale geometric morphisms.

See also

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Notes

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  1. ^ The assumptions that   is continuous and open imply that the set   is equal to the union of all open subsets   of   such that the restriction   is an injective map.
  2. ^ Consider the continuous open surjection   defined by   The set   for this map is the empty set; that is, there does not exist any non-empty open subset   of   for which the restriction   is an injective map.
  3. ^ And even if the polynomial function is not an open map, then this theorem may nevertheless still be applied (possibly multiple times) to restrictions of the function to appropriately chosen subsets of the domain (based on consideration of the map's local minimums/maximums).

Citations

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  1. ^ Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

References

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