Transcription of Section 13. Basis for a Topology - East Tennessee State ...
1 13. Basis for a Topology1 Section for a this Section , we consider a Basis for a Topology on a set which is, ina sense, analogous to the Basis for a vector space. Whereas a Basis for a vectorspace is a set of vectors which (efficiently; , linearly independently) generatesthe whole space through the process of raking linear combinations, a Basis for atopology is a collection of open sets which generates all open sets ( , elements ofthe Topology ) through the process of taking unions (see Lemma ). a set. Abasisfor a Topology onXis a collectionBof subsetsofX(calledbasis elements) such that(1)For eachx X, there is at least one Basis elementB Bsuch thatx B.
2 (2)Ifx Ba B2whereB1, B2 Bthen there isB3 Bsuch thatx B3andB3 B2 topologyTgenerated byBis defined as: A subsetU Xis inTif for eachx Uthere isB Bsuch thatx BandB U. (Therefore each Basis elementis inT.) need to prove that the alleged Topology generated by basisBis reallyin fact a Basis for a Topology2 Theorem a Basis for a Topology onX. DefineT={U X|x Uimpliesx B Ufor someB B},the Topology generated beB. ThenTis in fact a Topology set of real numbers (under the standard Topology ) is open ifandonly if it is a countable disjoint union of open intervals. This is one of the mostimportant results from Analysis 1 (MATH 4217/5217)!
3 A largely self-containedproof of this (only requiring a knowledge of lub and glb of a set of real numbers)can be found in my supplemental notes to Analysis 1 at: a Basis for the standard Topology onRis given by the set of all open intervalsof real numbers:B={(a, b)|a, b R, a < b} {( , b)|b R} {(a, )|a R}.In fact, a countable Basis for the standard Topology is givenbyB ={(a, b)|a, b Q, a < b}. This is based in part on the fact that a countable union of countablesets is countable (see Munkres Theorem ). See Exercise (a).Example Basis for the standard Topology onR2is given by the set of allcircular regions inR2:B={B((x0, y0), r)|r >0 andB((x0, y0), r) ={(x, y) R2|(x x0)2+(y y0)2< r2}}.
4 In fact, a countable Basis is similarly given by consideringallB((p0, q0), r) wherep0, q0 Qandr Qwherer > Basis for a Topology3 Example Basis for the standard Topology onR2is also given by the set ofall open rectangular regions inR2(see Figure on page 78).Example any set,B={{x}|x X}is a Basis for the discrete following result makes it more clear as to how a Basis can be used tobuild all open sets in a a set and letBbe a Basis for a topologyTonX. ThenTequals the collection of all unions of elements previous result allows us to create ( generate ) a Topology from a following result allows us to test a collection of open sets to see if it is a basisfor a given (X,T) be a topological space.
5 Suppose thatCis a collection ofopen sets ofXsuch that for each open subsetU Xand eachx U, there is anelementC Csuch thatx C U. ThenCis a Basis for the following lemma allows us to potentially compare the fineness/coarsenessto two topologies on setXbased on properties of respective Basis for a Topology4 Lemma be bases for topologiesTandT , respectively, the following are equivalent:(1)T is finer thanT.(2)For eachx Xand each Basis elementB Bcontainingx, there is a basiselementB Bsuch thatx B now define three topologies onR, one of which (the standard Topology )should already be familiar to the set of all open bounded intervals in the real line:B={(a, b)|a, b R, a < b}.
6 The Topology generated byBis thestandard be the set of all half open bounded intervals as follows:B ={[a, b)|a, b R, a < b}.The Topology generated byB is thelower limit topologyonR, denotedR`. {1/n|n N}. LetB ={(a, b)|a, b R, a < b} {(a, b)\K|a, b R, a < b}.The Topology generated byB is theK-topologyonR, Basis for a relationship between these three topologies onRis as given in topologies ofR`andRKare each strictly finer than the stan-dard Topology onR, but are not comparable with one a Topology on setXis a collection of subsets ofXwhose union equalsX. Thetopology generated by the subbasisSis defined to bethe collectionTof all unions of finite intersections of elements course we need to confirm that the Topology generated by a subbasis isin fact a a subbasis for a Topology onX.]
7 DefineTto be allunions of finite intersections of elements ofS. ThenTis a Topology : 5/28/2016
