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Planetmath: Topological Space (more info) Math for the people, by the people. Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS Advanced search topological space (Definition) "topological space" is owned by djao. [ full author list (2) ] (more info) Math for the people, by the people. Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS Advanced search compact (Definition) "compact" is owned by djao. [ full author list (2) ] Dense set 1 Dense set In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point x in X belongs to A or is a limit point of A.[1] Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A - for instance, every real number is either a rational number or has one arbitrarily close to it (see Diophantine approximation). Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A. Equivalently, A is dense in X if and only if the only closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A is empty. The density of a topological space X is the least cardinality of a dense subset of X. Density in metric spaces An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points), Then A is dense in X if Note that . If is a sequence of dense open sets in a complete metric space, X, then is also dense in X. This fact is one of the equivalent forms of the Baire category theorem. Examples The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets. By the Weierstrass approximation theorem, any given complex-valued continuous function defined on a closed interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space C[a,b] of continuous complex-valued functions on the interval [a,b], equipped with the supremum norm. Every metric space is dense in its completion. Properties Every topological space is dense in itself. For a set X equipped with the discrete topology the whole space is the only dense set. Every non-empty subset of a set X equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial. Denseness is transitive: Given three subsets A, B and C of a topological space X with A ⊆ B ⊆ C such that A is dense in B and B is dense in C (in the respective subspace topology) then A is also dense in C. The image of a dense subset under a surjective continuous function is again dense. The density of a topological space is a topological invariant. A topological space with a connected dense subset is necessarily connected itself. Dense set 2 Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuous functions f, g : X → Y into a Hausdorff space Y agree on a dense subset of X then they agree on all of X. Related notions A point x of a subset A of a topological space X is called a limit point of A (in X) if every neighbourhood of x also contains a point of A other than x itself, and an isolated point of A otherwise. A subset without isolated points is said to be dense-in-itself. A subset A of a topological space X is called nowhere dense (in X) if there is no neighborhood in X on which A is dense. Equivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure is empty. The interior of the complement of a nowhere dense set is always dense. The complement of a closed nowhere dense set is a dense open set. A topological space with a countable dense subset is called separable. A topological space is a Baire space if and only if the intersection of countably many dense open sets is always dense. A topological space is called resolvable if it is the union of two disjoint dense subsets. More generally, a topological space is called κ-resolvable if it contains κ pairwise disjoint dense sets. An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. A linear operator between topological vector spaces X and Y is said to be densely defined if its domain is a dense subset of X and if its range is contained within Y. See also continuous linear extension. A topological space X is hyperconnected if and only if every nonempty open set is dense in X. A topological space is submaximal if and only if every dense subset is open. References Notes [1] Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 048668735X General references • Nicolas Bourbaki (1989) [1971]. General Topology, Chapters 1–4. Elements of Mathematics. Springer-Verlag. ISBN 3-540-64241-2. • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR507446 Article Sources and Contributors 3 Article Sources and Contributors Dense set Source: http://en.wikipedia.org/w/index.php?oldid=456661398 Contributors: A19grey, Aleph4, Archelon, Austinmohr, Banus, Bdmy, Bluestarlight37, Calle, Can't sleep, clown will eat me, CiaPan, Cuaxdon, Denis.arnaud, Dino, ELLusKa 86, Erzbischof, Fly by Night, Giftlite, Graham87, Haemo, Headbomb, Isnow, Kae1is, Macrakis, Maksim-e, MathMartin, Michael Hardy, anonymous edits דניאל ב., דניאל צבי, Mks004, Nguyen Thanh Quang, Oleg Alexandrov, Physicistjedi, Poulpy, Tobias Bergemann, Utkwes, Vundicind, 25 License Creative Commons Attribution-Share Alike 3.0 Unported //creativecommons.org/licenses/by-sa/3.0/ Homeomorphism 1 Homeomorphism In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and 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 A continuous deformation between a coffee mug mathematical joke is that topologists can't tell their coffee cup from and a donut illustrating that they are homeomorphic. But there need not be a their donut,[1] since a sufficiently pliable donut could be reshaped to continuous deformation for two spaces to be the form of a coffee cup by creating a dimple and progressively homeomorphic—only a continuous mapping with enlarging it, while shrinking the hole into a handle. a continuous inverse. Topology is the study of those properties of objects that do not change when homeomorphisms are applied. As Henri Poincaré famously said, mathematics is not the study of objects, but instead, the relations (isomorphisms for instance) between them. Definition A function f: X → Y between two topological spaces (X, T ) and (Y, T ) is called a homeomorphism if it has the X Y following properties: • f is a bijection (one-to-one and onto), • f is continuous, • the inverse function f −1 is continuous (f is an open mapping). A function with these three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic. A self-homeomorphism is a homeomorphism of 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. Homeomorphism 2 Examples • The unit 2-disc D2 and the unit square in R2 are homeomorphic. • The open interval (a, b) is homeomorphic to the real numbers R for any a < b. • 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 R2 (a 2-dimensional plane). (Here we use A trefoil knot is homeomorphic to a circle. 2-sphere in the sense of a physical beach ball, not a circle or 4-ball.) Continuous mappings are not always realizable as • Let A be a commutative ring with unity and let S be a multiplicative deformations. Here the knot has been thickened subset of A. Then Spec(A ) is homeomorphic to {p ∈ Spec(A) : p ∩ to make the image understandable. S S = ∅}. • Rm and Rn are not homeomorphic for m ≠ n. • The Euclidean real line is not homeomorphic to the unit circle as a subspace of R2 as the unit circle is compact as a subspace of Euclidean R2 but the real line is not compact.
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