Homeomorphisms

If we give E the discrete topology and T the product topology, then T becomes a compact metric space homeomorphic to the Cantor set under the metric... 

A partial analogy between the dynamics of CA and the behaviors of continuous dynamical systems may be obtained by exploiting a fundamental property of CA systems namely, continuity in the Cantor-set Topology. We recall from section 2.2.1 that the collection of all one-dimensional configurations, or the CA phase space, r = where E = 0,1,..., fc 9 cr and Z is the set of integers by which each site of the lattice is indexed, is a compact metric space homeomorphic to the Cantor set under the metric... 

A series of four papers by G. W. Ford and others [ForG56,56a,56b, 57] amplified this work by using Polya s Theorem to enumerate a variety of graphs on both labelled and unlabelled vertices. These included connected graphs, stars (blocks) of given homeomorphic type, and star trees. In addition many asymptotic results were derived. The enumeration of series-parallel graphs followed in 1956 [CarL56], and in that and subsequent years Harary produced... 

A function cpis called a homeomorphism if it is bijective and both cpand its inverse cp -1 are continuous,... 

In one extreme case within the topological framework, the two objects can be brought into a perfect correspondence, demonstrating topological equivalence. In a more precise formulation, two topological spaces (Aj, T,) and (X2, T2) are called topologically equivalent or homeomorphic if there exists a function... 

Evidently, the topological space (X, T) is embedded in the compact topological space (X°o Tot), since (X, T) is homeomorphic to a subspace of (Xoo, Too), as it follows from the definitions given above. 

Sets of local coordinate systems describing certain local features of complicated objects are often advantageous when compared to a single, global coordinate system. Within a topological framework, the general theory of sets of local coordinate systems is called manifold theory. Often, the local coordinate systems are interrelated, and these relations can be expressed by continuous, and in the case of differentiable manifolds, by differentiable mappings, called homeomorphisms (see Equation (15)), and diffeomorphisms, respectively. 

A function cpis called a diffeomorphism if cp is a homeomorphism and both the function cpand its inverse cp1 are infinitely differentiable, that is, both cpand cp-1 belong to the class C00 of functions ... 

A space M where each point x e M has an open neighborhood homeomorphic to a set open within a Euclidean half-space Hn, is an K-dimensional manifold with boundary. 

Two graphs are said to be homeomorphic if one graph can be obtained from the other by... 

Several two-substrate mechanisms are homeomorphous, and lead to the general rate equation... 

Definition B.I Suppose that M and N are topological spaces, and suppose that f M —> N is a continuous function. Suppose m e M. Then f is a local homeomorphism at m if there is a neighborhood M containing m such that f M is invertible and its inverse is continuous. If f is a local homeomorphism at each m e M, then f is a local homeomorphism. 

Theorem B.I Suppose X, Y and Z are topological spaces. Suppose tt T —> X is afinite-to-one local homeomorphism. Suppose Z is connected and simply connected. Suppose f Z X is continuous. Then there is a continuous function f Z T such that f = n o f. 

Hall, G. R. 1984 Resonance zones in two-parameter families of circle homeomorphisms. SIAM Jl Math. Anal. 15(6), 1075-1081. 

Take the sphere and attach p handles to form the orientable surface Sp. The corresponding Euler characteristic is given by x(Sp) = 2 — 2p. For example, the torus is homeomorphic to a sphere with one handle and has x = 0. 

Take the sphere and attach q crosscaps to form the non-orientable surface Nq which has the Euler characteristic x(Nq) = 2 — q. For example the projective plane is homeomorphic to the sphere with one crosscap and has... 

Table 2 lists all thirty faces of the embedding in Fig. 3. Once the number of faces is known, the Euler genus can easily be verified using the Euler equation. In this case, we find an Euler genus of seven, which means that the surface is indeed homeomorphic to a sphere with seven crosscaps. 

Topological stereoisomers have identical bond connectivity, but they cannot be interconverted one into another by continuous deformation [51]. They are homeomorphic, not isotopic. Their molecular graphs are non-planar, either intrinsically or extrinsically. 

Extrinsically non-planar graphs are non-planar by virtue of their embedding in the 3D-space. They may be intrinsically planar at the same time (Fig. 6) this is the case of links (45) which are homeomorphic to disjoint sets of circuit graphs [71,72] (46) and knots (47a and b) which are homeomorphic to a circuit graph (48). 

Fig. 6. Extrinsically non-planar homeomorphic figures a) a link (45) and the pair of disconnected circuit graphs (46) b) the trefoil knot shown as its mirror-images 47a and 47b...

Proof. Let S = X t u u Xm be the decomposition into irreducible components. We know Xt is not contained in any one other X, and hence by irredudbility is not contained in their union. Thus there is a point x in Xj contained in just one of the irreducible components. But any point g is the image of x under the homeomorphism y gx 1y thus each point is in just one irreducible component. That is, the irreducible components are disjoint, and hence they equal the connected components. 

If A is k[S] for some Sslt", the definition makes S homeomorphic to its image as a subset of Spec A. Furthermore, the image is dense for if a closed set Z(I) contains S, each fin I vanishes at all points of S, so I = 0. As in (5.1), it follows that Spec A is irreducible iff S is irreducible. Simple topology also shows that Spec A is connected if S is. The converse of this is not true, and the last section shows that we don t want it to be true. 

The topological space Spec A is not a sufficiently complicated geometrical object to capture the full structure of A, since the topology is so weak. Indeed, for a field k, all the spaces Spec k[X, Y]/ f (X, Y) for irreducible/are homeomorphic. Consequently one tries to add more structure while still keeping a geometric flavor. 

For this we return to thinking of A as in some sense functions on Spec A. The open set Spec A Z(f) is canonically homeomorphic to the spectrum of the localized ring <4[/ 1] = Af, so it is reasonable to consider Af as the functions on that open set. Intuitively we are just allowing rational functions on the set where the denominator does not vanish. One can show that these functions have a reasonable local-determination property a function on a large open set U = (J U, is precisely determined by a family of functions /, on Ua in which/, and ff agree on Ua n Ufi. This says we have a sheaf of functions (see (15.6)). 

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