# prove quotient topology is a topology

If all Ui are balanced then the inequality f (sx) ≤ f(x) for all unit scalars s is proved similarly. A subset E of a topological vector space X is bounded if for every neighborhood V of 0, then E ⊆ tV when t is sufficiently large. The quotient space X/~ together with the quotient map q : X → X/~ is characterized by the following universal property: if g : X → Z is a continuous map such that a ~ b implies g(a) = g(b) for all a and b in X, then there exists a unique continuous map f : X/~ → Z such that g = f ∘ q. : × X → X are continuous functions (where the domains of these functions are endowed with product topologies). Remark 2.7 : Note that the co-countable topology is ner than the co- nite topology. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Then, we show that if Y is equipped with any topology having the universal property, then that topology must be the subspace topology. The Universal Property of the Quotient Topology It’s time to boost the material in the last section from sets to topological spaces. A map f:X→Y{\displaystyle f:X\to Y} is a quotient map (sometimes called an identification map) if it is surjective, and a subset U of Y is open if and only if f−1(U){\displaystyle f^{-1}(U)} is open. Let X be a topological vector space. In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. This is called the natural string of U The following result is the most important tool for working with quotient topologies. This implies that every Hausdorff topological vector space is Tychonoff. Let be a non-discrete locally compact topological field, for example the real or complex numbers. (6.48) For the converse, if $$G$$ is continuous then $$F=G\circ q$$ is continuous because $$q$$ is continuous and compositions of continuous maps are continuous. The sequence U• is/is a:. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. Let ˝ Y be the subspace topology on Y. This is not true for TVSs in general. Moreover, if a vector space X has countable dimension then every string contains an absolutely convex string. The vector space operation of addition is uniformly continuous and an open map. And if this vector topology on X has a neighborhood of 0 that is properly contained in X, then the continuity of scalar multiplication × X → X at the origin forces the existence of an open neighborhood of the origin in X that doesn't contain any "unbounded sequence". Each set in the sequence U• is called a knot of U• and for every index i, Ui is called the ith knot of U•.  That is, if x and y are points in X, and Nx is the set of all neighborhoods that contain x, and Ny is the set of all neighborhoods that contain y, then x and y are "topologically indistinguishable" if and only if Nx = Ny. It is easy to construct examples of quotient maps that are neither open nor closed. If X is a non-trivial vector space (i.e. In fact, this is true for topological group, since the proof doesn't use the scalar multiplications. Such a topology is called a vector topology or a TVS topology on X. If all ni are distinct then we're done, otherwise pick distinct indices i < j such that ni = nj and construct m• = (m1, ⋅⋅⋅, mk-1) from n• by replacing ni with ni - 1 and deleting the jth element of n• (all other elements of n• are transferred to m• unchanged). One forms the adjunction space X ∪fY by taking the disjoint union of X and Y and identifying a with f(a) for all a in A. If X has an uncountable Hamel basis then f is not locally convex and not metrizable.. (In fact, 5.40.b shows that J is a topology regardless of whether π is surjective, but subjectivity of π is part of the definition of a quotient topology.) However, it is very important in applications because of its compactness properties (see Banach–Alaoglu theorem). , Let X be a vector space and let U• = (Ui)∞i=1 be a sequence of subsets of X. to, In a general TVS, the closed convex hull of a compact set may, The convex hull of a finite union of compact, A vector subspace of a TVS that is closed but not open is, The convex hull of a balanced (resp. Topics include: Topological space and continuous functions (bases, the product topology, the box topology, the subspace topology, the quotient topology, the metric topology), connectedness (path This means that if s ≠ 0 then the linear map X → X defined by x ↦ s x is a homeomorphism. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. This permits the following construction: given a topological vector space X (that is probably not Hausdorff), … Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological union of those subspaces. Let X be a topological space and let C = {C α : α ∈ A} be a family of subsets of X with subspace topology. ∎. By definition of quotient topology, this condition is equivalent to that all cosets of Q in R are open.  Every linear map from (X, τf) into another TVS is necessarily continuous. Solution: False. Any vector topology on X will be translation invariant and invariant under non-zero scalar multiplication, and for every 0 ≠ x ∈ X, the map Mx : → X given by Mx (s) := s x is a continuous linear bijection. A compact subset of a TVS (not necessarily Hausdorff) is complete. A function h is a homeomorphism, and objects X and Y are said to be homeomorphic, if and only if the function satisfies the following conditions. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. This will soon be enhanced to more than a set-theoretic bijection (giving the “right” topology on R/Z). b.Is the map ˇ always an open map? The reason for this name is the following: if (fn) is a sequence of elements in X, then fn has limit f ∈ X if and only if fn(x) has limit f(x) for every real number x. A TVS homomorphism or topological homomorphism is a continuous linear map u : X → Y between topological vector spaces (TVSs) such that the induced map u : X → Im u is an open mapping when Im u, which is the range or image of u, is given the subspace topology induced by Y. Relate it to a quotient map a Hausdor space be studied instead of X all Knots of topological... Existence of a continuous dual space—the set X * of all strings in a complete,! Surjective and is the branch of topology that is also the finest vector topology is quotient... 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