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(Redirected from Homeomorphic)
:'''Topological equivalence' redirects here; see also topological equivalence (dynamical systems).''
In the mathematical field of topology, a 'homeomorphism' or 'topological isomorphism' (from the Greek words ''homoios'' = ''similar'' and ''μορφή (morphē)'' = shape = form (Latin deformation of morphe)) is a special isomorphism between topological spaces which respects topological properties. Two spaces with a homeomorphism between them are called 'homeomorphic'. From a topological viewpoint they are the same.
Roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated joke is that topologists can't tell the coffee cup from which they are drinking from the donut they are eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
Intuitively, a homeomorphism maps points in the first object that are "close together" to points in the second object that are close together, and points in the first object that are not close together to points in the second object that are not close together. Topology is the study of those properties of objects that do not change when homeomorphisms are applied.
A function ''f'' between two topological spaces ''X'' and ''Y'' is called a 'homeomorphism' if it has the following properties:
★ ''f'' is a bijection (1-1 and onto),
★ ''f'' is continuous,
★ the inverse function ''f'' −1 is continuous (f is an open mapping).
If such a function exists, we say ''X'' and ''Y'' are 'homeomorphic'. A 'self-homeomorphism' is a homeomorphism between a topological space and itself. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called 'homeomorphism classes'.
★ The unit 2-disc D2 and the unit square in 'R'2 are homeomorphic.
★ The open interval (−1, 1) is homeomorphic to the real numbers 'R'.
★ The product space S1 × S1 and the two-dimensional torus are homeomorphic.
★ Every uniform isomorphism and isometric isomorphism is a homeomorphism.
★ Any 2-sphere with a single point removed is homeomorphic to the set of all points in 'R'2 (a 2-dimensional plane).
The third requirement, that ''f'' −1 be continuous, is essential. Consider for instance the function ''f'' :[0, 2π) → S1 defined by ''f''(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism.
Homeomorphisms are the isomorphisms in the category of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms ''X'' → ''X'' forms a group, called the 'homeomorphism group' of ''X'', often denoted Homeo(''X'').
For some purposes, the homeomorphism group happens to be too big, but
by means of the isotopy relation, one can reduce this group to the
mapping class group.
★ Two homeomorphic spaces share the same topological properties. For example, if one of them is compact, then the other is as well; if one of them is connected, then the other is as well; if one of them is Hausdorff, then the other is as well; their homology groups will coincide. Note however that this does not extend to properties defined via a metric; there are metric spaces which are homeomorphic even though one of them is complete and the other is not.
★ A homeomorphism is simultaneously an open mapping and a closed mapping, that is it maps open sets to open sets and closed sets to closed sets.
★ Every self-homeomorphism in can be extended to a self-homeomorphism of the whole disk (Alexander's Trick).
The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly — it may not be obvious from the description above that deforming a line segment to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts.
This characterization of a homeomorphism often leads to confusion with the concept of homotopy, which is actually ''defined'' as a continuous deformation, but from one ''function'' to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space ''X'' correspond to which points on ''Y'' — one just follows them as ''X'' deforms. In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: homotopy equivalence.
There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an isotopy between the identity map on ''X'' and the homeomorphism from ''X'' to ''Y''.
★ Local homeomorphism
★ Diffeomorphism
★ Uniform isomorphism is an isomorphism between uniform spaces
★ Isometric isomorphism is an isomorphism between metric spaces
★ Dehn twist
★ Homeomorphism (graph theory) (closely related to graph subdivision)
★ Isotopy
★ Mapping class group
★
:'''Topological equivalence' redirects here; see also topological equivalence (dynamical systems).''
A continuous deformation between a coffee mug and a donut illustrating that they are homeomorphic. But there does not need to be a continuous deformation for two spaces to be homeomorphic.
In the mathematical field of topology, a 'homeomorphism' or 'topological isomorphism' (from the Greek words ''homoios'' = ''similar'' and ''μορφή (morphē)'' = shape = form (Latin deformation of morphe)) is a special isomorphism between topological spaces which respects topological properties. Two spaces with a homeomorphism between them are called 'homeomorphic'. From a topological viewpoint they are the same.
Roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated joke is that topologists can't tell the coffee cup from which they are drinking from the donut they are eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
Intuitively, a homeomorphism maps points in the first object that are "close together" to points in the second object that are close together, and points in the first object that are not close together to points in the second object that are not close together. Topology is the study of those properties of objects that do not change when homeomorphisms are applied.
| Contents |
| Definition |
| Examples |
| Notes |
| Properties |
| Informal discussion |
| See also |
| External links |
Definition
A function ''f'' between two topological spaces ''X'' and ''Y'' is called a 'homeomorphism' if it has the following properties:
★ ''f'' is a bijection (1-1 and onto),
★ ''f'' is continuous,
★ the inverse function ''f'' −1 is continuous (f is an open mapping).
If such a function exists, we say ''X'' and ''Y'' are 'homeomorphic'. A 'self-homeomorphism' is a homeomorphism between a topological space and itself. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called 'homeomorphism classes'.
Examples
A trefoil knot is homeomorphic to a torus. While this may seem illogical, in four dimensions they can easily be deformed continuously.
★ The unit 2-disc D2 and the unit square in 'R'2 are homeomorphic.
★ The open interval (−1, 1) is homeomorphic to the real numbers 'R'.
★ The product space S1 × S1 and the two-dimensional torus are homeomorphic.
★ Every uniform isomorphism and isometric isomorphism is a homeomorphism.
★ Any 2-sphere with a single point removed is homeomorphic to the set of all points in 'R'2 (a 2-dimensional plane).
Notes
The third requirement, that ''f'' −1 be continuous, is essential. Consider for instance the function ''f'' :
Homeomorphisms are the isomorphisms in the category of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms ''X'' → ''X'' forms a group, called the 'homeomorphism group' of ''X'', often denoted Homeo(''X'').
For some purposes, the homeomorphism group happens to be too big, but
by means of the isotopy relation, one can reduce this group to the
mapping class group.
Properties
★ Two homeomorphic spaces share the same topological properties. For example, if one of them is compact, then the other is as well; if one of them is connected, then the other is as well; if one of them is Hausdorff, then the other is as well; their homology groups will coincide. Note however that this does not extend to properties defined via a metric; there are metric spaces which are homeomorphic even though one of them is complete and the other is not.
★ A homeomorphism is simultaneously an open mapping and a closed mapping, that is it maps open sets to open sets and closed sets to closed sets.
★ Every self-homeomorphism in can be extended to a self-homeomorphism of the whole disk (Alexander's Trick).
Informal discussion
The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly — it may not be obvious from the description above that deforming a line segment to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts.
This characterization of a homeomorphism often leads to confusion with the concept of homotopy, which is actually ''defined'' as a continuous deformation, but from one ''function'' to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space ''X'' correspond to which points on ''Y'' — one just follows them as ''X'' deforms. In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: homotopy equivalence.
There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an isotopy between the identity map on ''X'' and the homeomorphism from ''X'' to ''Y''.
See also
★ Local homeomorphism
★ Diffeomorphism
★ Uniform isomorphism is an isomorphism between uniform spaces
★ Isometric isomorphism is an isomorphism between metric spaces
★ Dehn twist
★ Homeomorphism (graph theory) (closely related to graph subdivision)
★ Isotopy
★ Mapping class group
External links
★
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