Topological Spaces

If a set X is provided with a topology T, then the pair (X, T) is called a topological space. 

If the topology T is chosen as the metric topology, that is, if the T-open sets are precisely those which are open in some metric d introduced into the set X, then one obtains the metric topological space (X, T). Note that the metric topological space (X, T) is a Hausdorff space and also a normal space. 

Since the specification of topologies implies that all open sets are defined, the concept of continuity can also be generalized to topological spaces, even if distance functions are not given. 

Consider two topological spaces, (AT, T ) and (X2, T2), and a function cp from AT to X2. This function cpis continuous if and only if the inverse image of every T2-open set of AT is Ti-open in Xv. 

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... 

Connectedness is defined indirectly as the lack of disconnectedness a topological space (X, T) is connected if it is not disconnected. A connected open subset is often called a domain. 

The family F is called a finite cover if F contains only a finite number of F, subsets. If every open cover of a subset A of a topological space X contains a finite subcover, then the subset A of the topological space X is compact. The compactness property is a generalization of the elementary properties of closed and bounded intervals. 

Some non-compact topological spaces (X, T) can be converted into some compact topological spaces (Xx, Tx) by a technique called the Alexandrov one-point compactification. Here... 

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. 

THE EXPANDED DENATURED STATE AN ENSEMBLE OF CONFORMATIONS TRAPPED IN A LOCALLY ENCODED TOPOLOGICAL SPACE... 

For a locally compact topological space X, let Ftl X) denote the homology group of possibly infinite singular chains with locally finite support (Borel-Moore homology). The... 

Fig. 5.3 Principal component plots in pharmacophoric and topological spaces. Rectangles schematically indicate the activity zones .

Fig. 3. Schematic representation of the topological space of hydration water in silica fine-particle cluster (45). The processes responsible for the water spin-lattice relaxation behavior are restricted rotational diffusion about an axis normal to the local surface (y process), reorientations mediated by translational displacements on the length scale of a monomer (P process), reorientations mediated by translational displacements in the length scale of the clusters (a process), and exchange with free water as a cutoff limit.

Therefore, we can adopt as our fundamental hypothesis that the topological space under consideration (i.e., the vacuum) is described by 0(3) rather than U(l) and work out the consequences [11-20]. Some of the latter are reviewed in this chapter. An 0(3) group can also be formed by the complex unit vectors defined by... 

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. 

Basak, S. C., and G. D. Grunwald, Use of Topological Space and Property Space in Selecting Structural Analogs, in Mathematical Modelling and Scientific Computing, in press. 1998. 

We want to extend the concept of connectedness to group schemes more general than matrix groups. To do this we will associate with each of them some topological space. This space will usually have more points than just those of a closed set in k", and before we go on it is worth observing that even in our current material there are indications that we do not have all the points we should have. 

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 comparison, think of a differentiable or complex-analytic manifold. There again one has a topological space together with some additional structure and again one can describe the structure by a sheaf of functions, prescribing for each open set which functions are C°° or analytic. Thus X = Spec A with our sheaf on it is a sort of geometric object, and obviously... 

Show that an irreducible topological space containing more than one point cannot be HausdorfT. 

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