Math 396. Product topology - Department of Mathematics - Stanford

Math 396. Product topology - Department of Mathematics - Stanford

6 Pages · 2005 · 120 KB · English

Math 396. Product topology The aim of this handout is to address two points: metrizability of nite products of metric spaces, and the abstract characterization of the

Math 396. Product topology - Department of Mathematics - Stanford free download


Math 396 Product topology The aim of this handout is to address two points: metrizability of nite products of metric spaces, and the abstract characterization of the product topology in terms of universal mapping properties among topological spaces This latter issue is related to explaining why the de nition of the product topology is not merely ad hocbut in a sense the ight" de nition In particular, when you study topology more systematically and encounter the problem of topologizing in nite products of topological spaces, if you think in terms of the universal property to be discussed below then you will inexorably be led to the right de nition of the product topology for a product of in nitely many topological spaces (it is not what one would naively expect it to be, based on experience with the case of nite products) 1Metrics on finite products Let X 1; : : : ; X dbe metrizable topological spaces The product set X=X 1     X d admits a natural product topology, as discussed in class It is natural to ask if, upon choosing metrics  j inducing the given topology on each X j, we can de ne a metric on Xin terms of the  j's such that induces the product topology on X The basic idea is to nd a metric which describes the idea of \coordinatewise closeness", but several natural candidates leap out, none of which are evidently better than any others: max ((x 1; : : : ; x d) ; (x 0 1 ; : : : ; x 0 d )) = max 1 j d j( x j; x 0 j )  Euc ((x 1; : : : ; x d) ; (x 0 1 ; : : : ; x 0 d )) = v u u t d X j =1  j( x j; x 0 j ) 2  1(( x 1; : : : ; x d) ; (x 0 1 ; : : : ; x 0 d )) = d X j =1  j( x j; x 0 j )  p(( x 1; : : : ; x d) ; (x 0 1 ; : : : ; x 0 d )) = 0 @ d X j =1  j( x j; x 0 j )p 1 A 1 =p ; p 1 When X j= R for all j, with  j the usual absolute value metric, these recover the various concrete norms we've seen on X=Rd Our rst aim will be to show that all of these rather di erentlooking metrics are at least bounded above and below by a positive multiple of each other (which is the best we can expect, since they sure aren't literally the same), and so in particular they all de ne the same topology In fact, we will see that the common topology they de ne is the product topology We rst axiomatize the preceding examples Let N:R d ! Rbe any norm which satis es the property that on the orthant [0 ;1 )d with nonnegative coordinates it is a monotonically increasing function in each individual coordinate when all others are held xed Examples of such N's include our old friends k  kmax; k  k Euc; k  k 1; k  k p(for p 1) where we recall that k(a 1; : : : ; a n) k p = 0 @ d X j =1 j a jjp 1 A 1 =p : 1 2 Here is the general theorem which shows that many metrics (including all those mentioned above) on a product space are bounded above and below by a positive multiple of each other and hence determine the same theory of open sets, closed sets, and convergence of sequences Theorem 11 LetN:R d ! Rbe any norm as considered above Then for metric spaces (X j;  j) for 1 j d, with product space X=X 1     X d, the function  N : X X ! Rde ned by  N (( x 1; : : : ; x d) ; (x 0 1 ; : : : ; x 0 d )) = N( 1( x 1; x 0 1 ) ; : : : ;  d( x d; x 0 d )) is a metric on X, and al l such  N 's are bounded above and below by a positive multiple of each other Proof Let's rst check that each  N really is a metric Since Nis

------------- Read More -------------

Download math-396-product-topology-department-of-mathematics-stanford.pdf

Math 396. Product topology - Department of Mathematics - Stanford related documents

DEPARTMENT of HEALTH and HUMAN - Centers for Disease Control and

507 Pages · 2008 · 6.61 MB · English

influenza, natural disasters, and terrorism, while remaining focused on the threats to health and local, tribal and territorial health network.

A Typology of Victim Characterization in Television Crime Dramas

33 Pages · 2010 · 278 KB · English

her analysis of one season of Law & Order, NYPD Blue, and The Practice. She found that only

List of Developing Nations Afghanistan Albania Algeria Angola

2 Pages · 2011 · 538 KB ·

Algeria. Angola. Antigua and Barbuda. Argentina. Armenia. Azerbaijan Hungary. India. Indonesia. Iran, Islamic Republic of. Iraq. Jamaica. Jordan.

22 NAVAJO NATION COUNCIL | Office of the Speaker

2 Pages · 2013 · 295 KB · English

Law and Order Committee receives update regarding and an additional amount of $1.4 million to ensure operation through operations through the winter season.

Building Permits Granted Development Services Department City of San Antonio

84 Pages · 2012 · 272 KB · English

438 RICHLAND HILLS DR BLDG 10. DL CAMBRIDGE DEV GROUP, INC. (713)961-1336 x. 2251200. NEW 2-STORY MULTI-FAMILY APARTMEN. $947,363.00 2284202. 20x4=80 sq ft at csw, 171 sq ft at approach. $0.00. 3106 PIEDRA DE RIO. PRESIDIO CONST LLC. (210)679-8837 x. 2284203.

Department of History Postgraduate Handbook 2017-18

48 Pages · 2017 · 906 KB · English

Social and cultural change in early modern Ireland; the diffusion of print and the changing experience of . support for their modules (https://www.maynoothuniversity.ie/current-students). Social Media. The Department of History has a presence on social Format (e.g., film, video, DVD), that is, the

An integrated approach to product design and process selection

48 Pages · 2011 · 2.15 MB ·

Narayan Raman .. M? < Bs% .. a geometric series given by TEMP(y) = r * TEMP(

constraints facing the implementation of the greater new orleans urban water plan

5 Pages · 2015 · 480 KB · English

IMPLEMENTATION OF THE GREATER. NEW ORLEANS URBAN WATER PLAN. Annabel Visschedijk en Frans van de Ven*. On September 6th of last year the Greater New Orleans Urban Water Plan. (UWP) was presented. A comprehensive plan which addresses flooding caused by heavy rainfall and 

Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut

12 Pages · 2010 · 187 KB ·

between its embedded volume and minimum 3-way cut. and 5 we solve the 3-terminal case, giving matching up- cut of any embedded graph.

Litigation Boutique of the Year

4 Pages · 2009 · 2.28 MB ·

arenas. their clients are banks, accounting firms, airlines, energy companies, tobacco is not a cliché. it's gibbs & bruns's lockstep compensation, bredhoff & kaiser's . dollar litigation arising from the bankruptcy of TOUSA, Inc. The