A commonly used example is the compactopen topology, e. X y is continuous if and only if its components p 1 f, p 2 fare continuous. But th e stronger result that x is continuous is eaeily proved using the definition of the compactopen topology. In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology. In this paper we introduce the product topology of an arbitrary number of topological spaces. Let x be a topological space and z, y be subspaces such that z y x.
Topology underlies all of analysis, and especially certain large spaces such as the dual of l1z lead to topologies that cannot be described by metrics. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only. A metric space is a set x where we have a notion of distance. Notes on categories, the subspace topology and the product topology. Topological spaces form the broadest regime in which the notion of a continuous function makes sense. Y, from a topological space xto a topological space y, to be continuous. Both trace and product topology are characterized as being.
The cartesian product a b read a cross b of two sets a and b is defined as the set of all ordered pairs a, b where a is a member of a and b is a member of b. The quotient topology on xris the nest topology for which qis continuous. The open subsets of a discrete space include all the subsets of the underlying set. These are lecture notes for a four hour advanced course on general topology. In particular, if one considers the space x r i of all real valued functions on i, convergence in the product topology is the same as pointwise convergence of functions. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. The fine topology is one such function space topology see for example 5,8. X with x 6 y there exist open sets u containing x and v containing y such that u t v 3. Ifx,d is a metric space, then the space xn is metrizablewith respect to product topology. Functions or maps are fundamental to all of mathematics. They are useful in physics, however, because we can never measure a quantity at an exact position in space or time. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to. An important theorem about the product topology is tychonoffs theorem. The introduction of a new function space topology, called the graph topology, enabled him to tackle almost continuous functions.
Product of two spaces let x1 and x2 be two topological spaces. Let xbe a metric space with distance function d, and let abe a subset of x. That is to say, a subset u xris open if and only q 1u is open. In topology, one may attempt to put a topology on the space of continuous functions from a topological space x to another one y, with utility depending on the nature of the spaces. If both the function f and the inverse function f1. These notes describe three topologies that can be placed on the set of all functions from a set x to a space y. A set x with a topology tis called a topological space. Notes on categories, the subspace topology and the product topology john terilla fall 2014 contents 1 introduction 1 2 a little category theory 1 3 the subspace topology 3. Disc s \undersetn \in \mathbbz\prod discs which are not open subsets in the tychonoff topology.
If sis any set with the discrete topology, then any function f. Function spaces the same is they di er by a function of zero length. Suppose now that you have a space x and an equivalence relation you form the set of equivalence classes x. X are continuous, then f is called a homeomorphism. Any product of closed subsets of x i is a closed set in x. A subset v of xis said to be closed if xnv belongs to. In this paper five of these products are applied to problems on function spaces. X y, from a topological space x to a topological space y, to be continuous. We also prove a su cient condition for a space to be metrizable. My main issue is not being able to understand the what product space means and how is it different from the product topology. Introduction this week we will cover the topic of product spaces. If bis a basis for the topology of x and cis a basis for the topology of y. X y is a surjective function, then the quotient topology on y is the collection of subsets of y that have open inverse images under f. In particular, each r n has the product topology of n copies of r.
Then p 1 fand p 2 fare compositions of continuous functions, so they are both continuous. Compatibility of product topology with the vector space structure. Definition the product topology on uxl is the coarest topology such that all projection maps pm are continuous. Lecture notes on topology for mat35004500 following j. In this context, this topology is also referred to as the topology of pointwise convergence. But another connection with the theory of continuous lattices lurks in. Examples of function spaces february 11, 2017 that is, the compatibility of these fragments is exactly the assertion that they t together to make a function x. Haghnejad azer department of mathematics mohaghegh ardabili university p.
The introduction of a new function space topology, called the graph topology, enabled him to. But another connection with the theory of continuous lattices lurks in this approach to function spaces, which. This is shown in 10, and in that paper, the space h f r is compared to the product space squarer. A topological space is an aspace if the set u is closed under arbitrary intersections. The exponential law for function spaces with the compactopen topo logy is discussed in 1. Hence we need to see that there are subsets of the cartesian product set. Pdf topologies on spaces of continuous functions researchgate. I x t has two topology which are called product and box topology. Quotient spaces and quotient maps university of iowa. A band think of fas a rule that to any element a2a associates a unique object fa 2b. U form a subbasis for a topology on yx, known as the product topology. Traditionally, we draw x1 as a horizontal set, x2 as a vertical. Xis called a limit point of the set aprovided every open set ocontaining xalso contains at least one point a.
Show that x is hausdor if and only if is a closed subset of x x in the product topology. L, u i x open, then the topology generatedby it, is the coarsest topology containing subbasiss. Product topology the aim of this handout is to address two points. Let be a set and suppose there is a collection of topo logical spaces \. Suppose x is any topological space and y 1, 2 with the. Any subset of a hausdor space is itself a hausdor space with respect to the subspace topology. For each pair of topologies, determine whether one is a re. Closed sets, hausdorff spaces, and closure of a set. Let x be a set, let y be a topological space, let fn be a sequence in yx, and let f 2 yx. The product topology on x y is the topology having a basis bthat is the collection of all sets of the form u v, where u is open in xand v is open in y.
Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. They assume familiarity with the foundations of the subject, as taught in the twohour. Clearly these spaces are not for use when anything signi cant depends on the value of the function at any precise point. I from a topological space x to topological spaces yi separates points and closed sets if for every closed. The product topology is the smallest topology such that all the sets in f0 are open. Part iii topological spaces 10 topological space basics. Introduction to topology answers to the test questions stefan kohl. Note also that all sets in f 0are closed since they are complements of sets in f. In other words, the quotient topology is the finest topology on y for which f. Fu rstenberg consider n with the arithmetic progression topology. X n for all i, then the product or box topology on q a n is the same as the subset topology induced from q x n with the product or box topology. If x and y are metrisable, then the initial paragraphs of this article show that x. In other words, the quotient topology is the finest topology on y for which f is continuous. R with the usual topology is a compact topological space.
A quotient map has the property that the image of a saturated open set is open. Both trace and product topology are characterized as being the coarsest topology with respect to which certain maps inclusions, projections are. X and let a be a limitpoint of e then there exists a sequence a n n of e converges to a. Function spaces and product topologies on powers of spaces article in topology and its applications 15716. For every topological space z z and every function f.
The space rn is metrizable with respect to product topology. A on ais referred to as the subspace topology on a. Function spaces and product topologies 247 a similar argument shows that x is fccontinuous. As we have indicated, convergence in the product topology is the same as pointwise convergence of functions. Function spaces and product topologies on powers of spaces. Example 7 if y is a topological space, then the product y. For a metric space x, the space h f x of homeomorphisms on x with the fine topology is a topological group. The particular distance function must satisfy the following conditions. Introduction when we consider properties of a reasonable function, probably the. I am not able to understand how they are isomorphic.
Both the box and the product topologies behave well with respect to some properties. In particular the focus is on cx, the set of all continuous realvalued functions on xendowed with the topology of. It turns out that a great deal of what can be proven for. Notes on metric spaces these notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Topology of function spaces andrew marsh, phd university of pittsburgh, 2004 this dissertation is a study of the relationship between a topological space xand various higherorder objects that we can associate with x. Also available is the product topology on the space of set theoretic functions i. We leave it to the reader to prove that this is a topology.
U x y is open in the product topology if and only if, given any point x. Introduction to topology tomoo matsumura november 30, 2010 contents. The exponential law is valid also for the product z x d y of 4 and the topology of pointwise convergence on xt zxdy has the weak topology with respect to zxy, zxy. Notes on categories, the subspace topology and the product. A function from ato bis a subset f of a bsuch that for all ain athere is exactly one bin bsuch that a. Further, there is just one way to piece the fragments together. A topological space is an aspace if the set u is closed under. Product topology the aim of this handout is to address.