This space is homeomorphic to the product of a countable number of copies of the discrete space S.

The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval.

* Two compact Hausdorff spaces X < sub > 1 </ sub > and X < sub > 2 </ sub > are homeomorphic if and only if their rings of continuous real-valued functions C ( X < sub > 1 </ sub >) and C ( X < sub > 2 </ sub >) are isomorphic.

The unitary group U ( n ) is endowed with the relative topology as a subset of M < sub > n </ sub >( C ), the set of all n × n complex matrices, which is itself homeomorphic to a 2n < sup > 2 </ sup >- dimensional Euclidean space.

A Riemann surface X is a topological space that is locally homeomorphic to an open subset of C, the set of complex numbers.

For every x in X, the fiber over x is a discrete subset of C. On every connected component of X, the fibers are homeomorphic.

If X is connected, there is a discrete space F such that for every x in X the fiber over x is homeomorphic to F and, moreover, for every x in X there is a neighborhood U of x such that its full pre-image p < sup >− 1 </ sup >( U ) is homeomorphic to U x F. In particular, the cardinality of the fiber over x is equal to the cardinality of F and it is called the degree of the cover p: C → X.

#* More specifically, Hirsch showed there is an invariant set C that is homeomorphic to the ( N-1 )- dimensional simplexand is a global attractor of every point excluding the origin.

Then the mapping cone C < sub > f </ sub > is homeomorphic to two disks joined on their boundary, which is topologically the sphere S < sup > 2 </ sup >.

We require that for every x in E, there is an open neighborhood U ⊂ B of π ( x ) ( which will be called a trivializing neighborhood ) such that π < sup >− 1 </ sup >( U ) is homeomorphic to the product space U × F, in such a way that π carries over to the projection onto the first factor.

A manifold with boundary is similar, except that a point of is allowed to have a neighborhood that is homeomorphic to the half-space

Suppose that c is a simple closed curve in a closed, orientable surface S. Let A be a tubular neighborhood of c. Then A is an annulus and so is homeomorphic to the Cartesian product of

The knot complement X < sub > K </ sub > is a compact 3-manifold ; the boundary of X < sub > K </ sub > and the boundary of the neighborhood N are homeomorphic to a two-torus.

In higher dimensions, given any point, it has a neighborhood locally homeomorphic to, and one can apply the same procedure.

A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to the Euclidean space E < sup > n </ sup > ( or, equivalently, to some connected open subset of E < sup > n </ sup >).

An n-manifold will mean a topological manifold such that every point has a neighborhood homeomorphic to R < sup > n </ sup >.

By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of R < sup > n </ sup >.

* every point of M has a neighborhood homeomorphic to an open ball in R < sup > n </ sup >.

* every point of M has a neighborhood homeomorphic to R < sup > n </ sup > itself.

A Euclidean neighborhood homeomorphic to an open ball in R < sup > n </ sup > is called a Euclidean ball.

A topological manifold with boundary is a Hausdorff space in which every point has a neighborhood homeomorphic to an open subset of Euclidean half-space ( for a fixed n ):

This can mean simply that a neighborhood of each vertex ( i. e. the set of simplices that contain that point as a vertex ) is homeomorphic to a n-dimensional ball.

Furthermore, the theorem says that if two subsets U and V of R < sup > n </ sup > are homeomorphic, and U is open, then V must be open as well.

* Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set.

An example is given by the real numbers, which are complete but homeomorphic to the open interval ( 0, 1 ), which is not complete.

Another example is given by the irrational numbers, which are not complete as a subspace of the real numbers but are homeomorphic to N < sup > N </ sup > ( see the sequence example in Examples above ).

A continuous deformation between a coffee mug and a torus | donut illustrating that they are homeomorphic.

Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.

Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not.

If such a function exists, we say X and Y are homeomorphic.

* The unit 2-disc D < sup > 2 </ sup > and the unit square in R < sup > 2 </ sup > are homeomorphic.

* The product space S < sup > 1 </ sup > × S < sup > 1 </ sup > and the two-dimensional torus are homeomorphic.

Separable metrizable spaces can also be characterized as those spaces which are homeomorphic to a subspace of the Hilbert cube, i. e. the countably infinite product of the unit interval ( with its natural subspace topology from the reals ) with itself, endowed with the product topology.

A stronger assumption is necessary ; in dimensions four and higher there are simply connected manifolds which are not homeomorphic to an n-sphere.

Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence ( or, much more deeply, existence ) of mappings.

In 1961 Milnor disproved the Hauptvermutung by exhibiting two simplicial complexes which are homeomorphic but combinatorially distinct.

Every homeomorphism is a homotopy equivalence, but the converse is not true: for example, a solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent.

Infinite volume manifolds can have many different types of geometric structure: for example, R < sup > 3 </ sup > can have 6 of the different geometric structures listed above, as 6 of the 8 model geometries are homeomorphic to it.

The genus classifies classifies compact Riemann surfaces up to homeomorphism, i. e., two such surfaces are homeomorphic ( but not necessarily diffeomorphic ) if and only if their genus is the same.

A topological space homeomorphic to a separable complete metric space is called a Polish space.

For compact space | compact 2-dimensional surfaces without boundary ( topology ) | boundary, if every loop can be continuously tightened to a point, then the surface is topologically Homeomorphism | homeomorphic to a 2-sphere ( usually just called a sphere ).

The conjecture states: An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.

In the process, he discovered some interesting examples of simply connected non-compact 3-manifolds not homeomorphic to R < sup > 3 </ sup >, the prototype of which is now called the Whitehead manifold.

In general, any topological space that is homeomorphic to the 3-sphere is called a topological 3-sphere.

An n-dimensional manifold ( either embedded in a finite dimensional vector space, or an abstract manifold ) is called non-orientable if it is possible to take the homeomorphic image of an n-dimensional ball in the manifold and move it through the manifold and back to itself, so that at the end of the path, the ball has been reflected, using the same definition as for surfaces above.

In algebraic topology a compact topological space which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a polyhedron ( see,, ).

A domain that covers an entire sequence is called the homeomorphic domain by PIR ( Protein Information Resource ).

# X is homeomorphic to the spectrum of a bounded distributive lattice L. In this case, L is isomorphic ( as a bounded lattice ) to the lattice K < sup ></ sup >( X ) ( this is called Stone representation of distributive lattices ).

A genus one handlebody is homeomorphic to B < sup > 2 </ sup > × S < sup > 1 </ sup > ( where S < sup > 1 </ sup > is the circle ) and is called a solid torus.

is then homeomorphic to its image in ( also with the subspace topology ) and is called a topological embedding.

* A Kleinian group is called topologically tame if it is finitely generated and its hyperbolic manifold is homeomorphic to the interior of a compact manifold with boundary.

* If the form is E < sub > 8 </ sub >, this gives a manifold called the E8 manifold, a manifold not homeomorphic to any simplicial complex.

A presentation of a topological manifold is a second countable Hausdorff space that is locally homeomorphic to a linear space, by a collection ( called an atlas ) of homeomorphisms called charts.