A study on compactness in metric spaces and topological spaces

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1 Pure and Appled Matheatcs Journal 24; 3(5): 5-2 Publshed onlne October 2, 24 ( do: 648/jpaj24353 ISSN: (Prnt); ISSN: (Onlne) A study on copactness n etrc spaces and topologcal spaces Rabeya Akter, Nour Mohaed Chowdhury 2, Mohaad Saf Ullah 3 Departent of Matheatcs, Jagannath Unversty, Dhaka, Bangladesh 2 Departent of Matheatcs, World Unversty of Bangladesh, Dhaka, Bangladesh 3 Departent of Matheatcs, Colla Unversty, Colla, Bangladesh Eal address: rabeyaju@galco (R Akter), ranapu6@galco, (N M Chowdhury), Safru985@galco (M S Ullah) To cte ths artcle: Rabeya Akter, Nour Mohaed Chowdhury, Mohaad Saf Ullah A Study on Copactness n Metrc Spaces and Topologcal Spaces Pure and Appled Matheatcs Journal Vol 3, No 5, 24, pp 5-2 do: 648/jpaj24353 Abstract: Topology ay be consdered as an abstract study of the lt pont concept As such, t stes n part fro recognton of the fact that any portant atheatcal topcs depend entrely upon the propertes of lt ponts Ths study shows that copactness ples lt pont copactness but not conversely and every copact space s locally copact but not conversely Ths study also shows that copactness, lt pont copactness and sequentally copactness are equvalent n etrzable spaces and the product of fntely any copact spaces s a locally copact space Ths study ntroduce t here as an nterestng applcaton of the Tychonoff theore Keywords: Metrc Spaces, Topologcal Space, Copact Space, Locally Copact Space, Sequentally Copactness, Neghborhood Introducton The theory of topologcal spaces or, as t s called, pont set or general topology, has becoe one of the eleentary buldng blocks underlyng dver s branches of atheatcs Its concepts and ethods have enrched nuerous other felds of atheatcs and gven consderable pulse to ther further developent For these reasons, topology counts aong those few basc structures, whch alone gve access to odern atheatcal research In ths work, we have tred to accuulate several types of concepts of separaton axos on topologcal space, obtaned fro dfferent texts Nuerous proofs, exaples and solved probles are ncluded n ths work Up to now, no work ever coplete We only try to dscuss the clear concepts of copactness on topologcal spaces, ts propertes and applcatons, so that one can see these aterals concernng copactness on topologcal spaces at one place Objectve The advantage n copact spaces s that one ay study the whole space by studyng a fnte nuber of open subsets We shall see ths when we prove that a contnuous functon f: X Y fro a copact etrc space X to a etrc space Y s unforly contnuous In concluson we shall exane soe copact surfaces that ay be fored by dentfyng edges of a rectangle 2 Procedure In the proposed study, frst, we shall gve soe necessary defntons and states soe necessary theores n order to present the paper n a self contaned anner Afterward, we shall prove the ost portant and popular theore naely Hene-Borel Theore Then we shall study locally copact spaces of a etrc space and n general of a topologcal space Next, we shall dscuss σ-copact and fnally copact spaces and soe of ther portant propertes Fnally, we shall study copletely copact spaces, ant-copact spaces and copletely dense subsets n a space and soe of ther portant propertes Let {G } be a class of subsets of X such that A G for soe A X The class {G } s then called a cover of A, and an open cover f each G s open Furtherore, f a fnte subclass of {G } s also a cover of A, e f G, G,, G {G } such that A 2

2 6 Rabeya Akter et al: A Study on Copactness n Metrc Spaces and Topologcal Spaces G G G, then {G } s sad to be 2 reducble to a fnte cover, or contans a fnte subcover 3 Copact Sets A subset A of a topologcal space (X, T) s copact f every open cover of A s reducble to a fnte subcover In other words, f A s copact and A G, where G are open sets, then one can select a fnte nuber of the open sets, say G, G,, G, so that A 2 G G G 2 4 Copact Spaces A topologcal space X s sad to be copact f for each open coverng (U α ) α I of X there s a fnte subcoverng (V β ) β J We ay alternatvely defne copactness by the stateent, X s copact f for each open coverng (U α ) α I of X there s a fnte subset of ndces {α, α 2,,α n } such that the collecton Uα, Uα,, U covers X 2 αn A subset C of a topologcal space X s sad to be copact, f C s a copact topologcal space n the relatve topology The real lne R s not copact, for the coverng of R by open ntervals ={(n, n+2) n Z } contans no fnte subcollecton that covers R Any space X contanng only fntely any ponts s necessarly copact, because n ths case every open coverng of X s fnte 2 Copactness Theore 2 [6] : A subset C of a topologcal space X s copact f and only f for each open coverng (U α ) α I of C, U α open n X, there s a fnte subcoverng U α,, U α n of C Let C be copact and let (U α ) α I be an open coverng of C Thus C α I U α, hence C= α I (U α C), so that the faly (U α C) α I s a coverng of the topologcal space C by relatvely open sets U α C Snce C s copact, there s a fnte collecton of ndces {α, α 2,,α n } such that U α C, U α 2 C,, U α n C covers C; that s C = n Consequently, C (U C) n U α = = Conversely, suppose that for each open coverng (U α ) α I of C, there s a fnte subcoverng U α, U α 2,, U α n of C We ust show that gven any coverng of the topologcal space C by a faly (V β ) β J of relatvely open subsets of C, there s a fnte subcoverng For each β J, Snce V β s relatvely open n C, there s an open subset U β α of X such that V β =U β C But C = β J V β, therefore C β JU β and (U β ) β J s a coverng of the subset C by open sets By our hypothess, there s a fnte subcoverng U β, U β,, U 2 β of C Thus, C U (( U = ) C = β (U C) = V = β = β = β and C = Hence the coverng (V β ) β J of C by relatvely open sets has a fnte subcoverng V β, V β 2,, V β Copactness ay be characterzed n ters of neghbourhoods Theore 22 [6] : A topologcal space X s copact f and only f, whenever for each x X a neghbourhood N x of x s gven, there s fnte nuber of ponts n x, x 2,, x n of X such that X= N x = Theore 23 [4] : Any closed subspace of a copact space s copact Let Y be a closed of a copact space X, and let {G } be an open cover of Y Each G, beng open n the relatve topology on Y, s the ntersecton wth Y of an open subset H of X Snce Y s closed, the class coposed of Y and all the H s s an open cover of X, and snce X s copact, ths open cover has a fnte subcover If Y occurs n ths subcover, we dscard t What reans s a fnte class of H s whose unon contans X Our concluson that Y s copact now follows fro the fact that the correspondng G s for a fnte subcover of the orgnal open cover of Y Theore 24 [4] : Any contnuous age of a copact space s copact Let f: X Y be a contnuous appng of a copact space X nto an arbtrary topologcal space Y We ust show that f(x) s a copact subspace of Y Let {G } be an open cover of f(x) As n the above proof, each G s the ntersecton wth f(x) of an open subset H of Y It s clear that {f (H )} s an open cover of X, and by the copactness of X t has a fnte subcover The unon of the fnte class of H s of whch these are the nverse ages clearly contans f(x), so the class of correspondng G s s a fnte subcover of the orgnal open cover of f(x), and f(x) s copact Theore 25 [3] : If A s a copact subset of a Hausdorff space X and x s a pont of X A, then there are dsjont neghbourhoods of x and of A Consequently each copact subset of a Hausdorff space s closed Snce X s Hausdorff there s a neghbourhood U of each pont y of A such that x does not belong to the closure U Because A s copact there s a fnte faly U, U,, U n of open sets coverng A such that x U for =,,, n If V = {U: =,,, n}, then A V and x V Consequently X V and V are dsjont neghbourhoods of x and A

3 Pure and Appled Matheatcs Journal 24; 3(5): Theore 26 [3] : Let f be a contnuous functon carryng the copact topologcal space X onto the topologcal space Y Then Y s copact, and f Y s Hausdorff and f s one to one then f s a hooorphs If α s an open cover of Y, then the faly of all sets of the for f [A] for A n α, s an open cover of X whch has a fnte subcover The faly of ages of ebers of the subcver s a fnte subfaly of α whch covers Y and consequently Y s copact Suppose that Y s hausdorff and f s one to one If A s a closed subset of X, then A s copact and hence ts age f[a] s copact and therefore closed Then (f ) [A] s closed for each closed set A and f s contnuous Theore 27 [3] : If A and B are dsjont copact subsets of a Hausdorff space X, then there are dsjont neghbourhoods of A and B Consequeltly each copact Hausdorff space s noral For each x n A there s a neghbourhood of x and a neghbourhood of B whch are dsjont Consequently there s a neghbourhood U of x whose closure s dsjont fro B, and snce A s copact there s a fnte faly U, U,,U n such that U s dsjont fro B for =,,n and A V = {U : =,,, n} Then v s a neghbourhood of A and X V s a neghborhood of B whch s dsjont fro V Theore 28 [3] : If X s a regular topologcal space, A a copact subset, and U a neghbourhood of A, then there s a closed neghbourhood V of A such that V U Consequently each copact regular space s noral Because X s regular, for each x n A there s an open neghbourhood W of x such that W U, and by copactness there s a fnte open cover W, W,, W n of A such that W U for each Then V = { W : =,,,n} s the requred neghbourhood of A Theore 29 [3] : If X s a copletely regular space, A s a copact subset and U s a neghbourhood of A, then there s a contnuous functon f on X to the closed nterval [, ] such that f s one on A and zero on X U Proposton 2 [9] : For a topologcal space X, the followng are equvalent () X s copact (2) If F s any faly of closed sets n X wth fp, then F =φ (3) Every net n X has a cluster pont Theore (The Hene-Borel Theore) 2 [4] : Every closed and bounded subspace of the real lne s copact A closed and bounded subspace of the real lne s a closed subspace of soe closed nterval [a, b], t suffces to show that [a, b] s copact If a=b, ths s clear, so we ay assue that a<b we know that the class of all ntervals of the for [a, d) and (c, b], where c and d are any real nubers such that a<c<b and a<d<b, s an open subbase for [a, b]; therefore the class of all [a, c] s and all [d, b] s s a closed subbase Let S={[a, c ], [d j, b]}be a class of these subbasc closed sets wth the fnte ntersecton property It suffces to show that the ntersecton of all sets n S s non-epty We ay assue that S s non-epty If S contans only ntervals of the type [a, c ], or only ntervals of the type [d j, b], then the ntersecton clearly contans a or b We ay thus assue that S contans ntervals of both types We now defne d by d =sup {d }, and we coplete the proof by showng that d c for every Suppose that c < d for soe Then by the defnton of d there exsts a d j such that c < d j Snce [a, c ] [d j, b] = φ, ths contradcts the fnte ntersecton property for S and concludes the proof Proposton 22 [9] : (The Tube Lea) Let X X 2 be a product space where X 2 s copact and let p X be fxed Let N be an nbd of the slce {p} X 2 n the product space Then there exsts an nbd U of P n X such that {p} X 2 U X 2 N (The nbd U X 2 s called a tubular nbd or sply a tube) Proposton 23 [9] : A fnte product space X= n s copact ff each factor X j j= space X j s copact Note: The above result holds for arbtrary products also The general result s known as the Tychonoff Theore Theore 2 [4] : (The Generalzed Hene-Borel Theore) Every closed and bounded subspace of R n s copact Theore 22 [2] : (Extree value theore) Let f: X Y be contnuous, where Y s an ordered set n the order topology If x s copact, then there ext ponts c and d n X such that f(c) f(x) f(d) for every x X The extree value theore of calculus s the specal case of ths theore that occurs when we take X to be a closed nterval n R and Y to be R A real nuber a> s called a Lebesgue nuber for our gven open cover {G } f each subset of X whose daeter s less than a s contaned n at least one G Let X be a topologcal space If (x n ) s a sequence of ponts of X, and f n <n 2 < <n < s an ncreasng sequence of postve ntegers, then the sequence (y ) defned by settng y =x n s called a subsequence of the sequence (x n ) The space X s sad to be sequentally copact f every sequence of ponts of X has a convergent subsequence Theore 23 [4] : (Lebesgue s Coverng Lea) In a sequentally copact etrc space, every open cover has a Lebesgue nuber A space X s sad to be lt pont copact f every nfnte subset of X has a lt pont Note: On hstorcal grounds, soe call t Frechet copactness ; others call t the Bolzano weerstrass property Theore 24 [2] : Copactness ples lt pont copactness, but not

4 8 Rabeya Akter et al: A Study on Copactness n Metrc Spaces and Topologcal Spaces conversely Exaple: Let Y consst of two ponts; gve Y the topology consstng of Y and the epty set Then the space X= Z + Y s lt pont copact, for every nonepty subset of X has a lt pont It s not copact, for the coverng of X by the open sets U n = {n} Y has no fnte subcollecton coverng X Now we prove that copactness and lt pont copactness are equvalent n atrzable spaces Theore 25 [2] : Let X be a etrzable space Then the followng are equvalent: X s copact 2 X s lt pont copact 3 X s sequentally copact Theore 26 [4] : Any contnuous appng of a copact etrc space nto a etrc space s unforly contnuous 3 Locally Copact Space A space X s sad to be locally copact at x f there s soe copact subspace C of X that contans a neghborhood of X f X s locally copact at each of ts ponts, X s sad sply to be locally copact Every copact space s locally copact, snce a topologcal space s always a neghborhood of each of ts ponts Every dscrete space s locally copact, snce for each pont p n t, the sngleton {p} s a copact nbd Every ndscrete space X s locally copact Consder the real lne R wth the usual topology Observe that each pont p R s nteror to a closed nterval, eg [p-δ, p+δ] and that the closed nterval s copact by the Hene- Borel theore Hence R s locally copact space On the other hand R s not a copact space For exaple, the class = {, (-3, -), (-2, ), (-, ), (, 2), (, 3), } s an open cover of R but contans no fnte subcover Therefore R s locally copact but not copact Slarly C and Z are locally copact but not copact Theore 3 [7] : Every closed subset of a locally copact space s locally copact Let B be a closed subset of a locally copact space X and let x be any pont of B Then there exsts a copact neghborhood C of x Therefore B C s a closed subset of C Therefore B C s a copact subset of C (snce every closed subset of a copact space s copact), and so B C s a copact subset of B Thus B C s a copact neghborhood of x n the subspace B of X Hence B s locally copact Theore 32 [7] : Every separated locally copact space s regular A subset B of a topologcal space X s sad to be locally copact f B s locally copact consdered as a subspace of X Theore 33 [7] : Let x be any pont of a regular locally copact space X Then the class consstng of all closed copact neghborhood of x s a local base at x The class consstng of all those open neghborhoods of x whose closures are copact s also a local base at x Theore 34 [7] : Let a subset B of a regular locally copact space X be the ntersecton of a closed and an open subset of X Then B s locally copact Let x be any pont of B Then there s a neghborhood N of x such that B N s a closed subset of the subspace N of X Snce X s regular and locally copact, then by the prevous theore there s a copact neghborhood K of x such that K N Snce B N s a closed subset of the subspace N, (B N) K s a closed subset of K Snce K s copact, (B N) K s copact Therefore (B N) K = B K s a copact neghborhood of x n the subspace B of X Hence B s locally copact Theore 35 [3] : If U s a neghborhood of a closed copact subset A of a regular locally copact topologcal space X, then there s a closed copact neghborhood V of A such that A V U Moreover, there s a contnuous functon f on X to the closed unt nterval such that f s zero on A and one on X V For each pont x of A there s a neghbourhood W whch s closed copact subset of U Snce A s copact t ay be covered by a fnte faly of such neghbourhoods and ther unon s a closed copact neghbourhood V of A Then V wth the relatve topology s a regular copact space whch s therefore noral Hence there s a contnuous functon g on V to the closed unt nterval such that g s zero on A and one on V V (V s the nteror of V) Let f equal g on V and one on X V Then f s contnuous because V and X V are separated and f s contnuous on V and X V Theore 36 [7] : A regular locally copact space s copletely regular Theore 37 [7] : A subset B of a locally copact space X s closed f B C s closed for every copact subset C of X Let B C be closed for every copact subset C of X Let x be any contact pont of B Snce X s locally copact, there exsts a copact neghbourhood C of x Now let N be any neghbourhood of x We know that, f x s any pont of topologcal space X, then the class v(x) of neghbourhoods of x If M and N are ebers of v(x) then M N s also a eber of v(x) Hence by the above theore C N s a neghbourhood of x,

5 Pure and Appled Matheatcs Journal 24; 3(5): and as such (C N) B s non vod Hence x s a contact pont of B C But B C s closed Also we know that a subset B of a topologcal space s closed f and only f B = B Hence fro the above x s n B C and therefore n B Thus B s closed Theore 38 [3] : If X s a locally copact topologcal space whch s ether Hausdorff or regular, then the faly of closed copact neghbourhoods of each pont s a base ts neghbourhood syste Let x be a pont of X, C a copact nehgbourhood of x, and U an arbtrary nehgbourhood of x If X s regular, then there s a closed nehgbourhood V of x whch s a subset of the ntersecton of U and the nteror of C, and evdently V s closed and copact If X s Hausdorff and W s the nteror of U C, then, snce W s a copact Hausdorff space, W contans a closed copact set V whch s a neghborhood of x n W ; but V s also a neghborhood of x n W (that s, wth respect to the relatvzed topology for W) and s therefore a neghborhood of x n X Note: In partcular t follows that every locally copact Hausdorff space s regular; actually a stronger stateent s true Proposton 3 [8] : An open subspace of a locally copact, regular space s locally copact Let X be locally copact and regular and suppose Y s open n X Let y Y Then Y s a neghborhood of y n X and so there exsts a copact neghborhood C of y such that C Y But then, Y s locally copact at y Snce y was an arbtrary pont of Y, t follows that Y s locally copact n the relatve topology on t Proposton 32 [8] : Let X be a Hausdorff space and Y be a dense subset of X If Y s locally copact n the relatve topology on t, then Y s open n X Theore 39 [8] : A subspace of a locally copact, Hausdorff space s locllay copact ff t s open n ts closure Let X be a locally copact Hausdorff space and Y be a subspace of X Put Z = Y Then Z tself s locally copact for t s easy to show that local copactness s weakly heredtary Also Z s Hausdorff and therefore regular (Snce every locally copact, Hausdorff space s regular) Now f Y s open n Z then Y s locally copact by proposton Conversely f Y s locally copact, then by the last proposton appled to Z, we see that Y s open n Z Theore 3 [] : The product of fntely any locally copact spaces s a locally copact space Let S, S 2,, and S n each be locally copact, and let p = (p,, p n ) be any pont n S S 2 S n In each space S there s an open set U contanng p and havng copact closure Then the bass eleent U U 2 U n contans p and has closure U U 2 U whch s copact by the n Tychonoff theore Hence S S 2 S n s locally copact Theore 3 [3] : If a product s locally copact then each coordnate space s locally copact and all except a fnte nuber of coordnate spaces are copact If a product s locally copact, then each coordnate space s locally copact because projecton nto a coordnate space s open We know that f nfntely any coordnate spaces are noncopact, then each copact subset of the product s nowhere dense, and no pont has a copact neghbourhood 4 Copactfcaton Let X be a locally copact Hausdorff space Take soe object outsde X, denoted by the sybol for convenence, and adjon t to X, forng the set Y = X { } Topologze Y by defnng, the collecton of open sets n Y to be all sets of the followng types: () U, where U s an open subset of X, (2) Y C, where C s copact subset of X The space Y s called the one-pont copactfcaton of X We need to check that ths collecton of open sets s a topology on Y The epty set s a set of type of (l), and the space Y s set of type (2) Checkng that the ntersecton of two open sets s open nvolves three cases: U U 2 s of the type (l) (Y C ) (Y C 2 ) = Y (C C 2 ) s of the type (2) U (Y C ) = U (X C ) s of the type (), because C s closed n X Slarly, one checks that the unon of any collecton of open sets s open: U U α = U s of the type () U (Y C β ) = Y ( C β ) = Y C s of type (2) (U U α ) (U (Y C β )) = U (Y C) = Y (C U) whch s of the type (2), because (C U) s a closed subset of C and therefore copact The one pont copactfcaton of the real lne R s hoeoorphc wth the crcle Slarly, the one-pont copactfcaton of R 2 s hoeorphc to the sphere S 2 Theore 4 [2] : Let X be a locally copact Hausdorff space, whch s not copact; Let Y be the one pont Copactfcaton of X Then Y s a copact Hausdorff space; X s a subspace of Y; the set Y X conssts of a sngle pont; and cl(x) = Y Theore 42 [3] : (ALEXANDROFF) The one-pont copactfcaton X* of a topologcal space X s copact and X s a subspace The space X* s Hausdorff

6 Rabeya Akter et al: A Study on Copactness n Metrc Spaces and Topologcal Spaces f and only f X s locally copact and Hausdorff A set U s open n X* ff (a) U X s open n X and (b) whenever U, then X U s copact Consequently fnte ntersectons and arbtrary unons of sets open n X* ntersect X n open sets If s a eber of the ntersecton of two open subsets of X *, then the copleent of the ntersecton s the unon of two closed copact subsets of X and s therefore closed and copact If belongs to the unon of the ebers of a faly of open subsets of X *, then belongs to soe eber U of the faly, and the copleent of the unon s a closed subset of the copact set X U and s therefore closed and copact Consequently X* s a topologcal space and X s a subspace If s an open cover of X*, then s a eber of soe U n and X U s copact, and hence there s a fnte subcover of U Therefore X* s copact If X* s a Hausdorff space, then ts open subspace X s a locally copact Hausdorff space Fnally t ust be shown that X* s a Hausdorff space f X s a locally copact Hausdorff space It s only necessary to show that, f x X, then there are dsjont neghborhoods of x and But snce X s locally copact and Hausdorff there s a closed copact neghbourhood U of x n X and X* U s the requred neghbourhood of Theore 43 [7] : A one-pont copacfcalon of a topologcal space A s separated ff A s separated and locally copact Lea 4 [2] : Let X be Hausdorff space Then X s locally copact at x f and only f for every neghbourhood U of x, there s a neghbourhood V of x such that cl(v) s copact and cl(v) U Corollary 4 [2] : Let X be a locally copact Hausdorff space and Y be a subspace of X If Y s closed n X or opens n X, then Y s locally copact Suppose that Y s closed n X Gven y Y, let C be a copact set n X contanng the neghbourhood U of x n X Then C Y s closed n C and thus copact Then t contans the neghbourhood U Y of y n Y (We have not used the Hausdorff condton here) Agan suppose that Y s open n X Gven y Y, we apply the prevous lea to choose a neghbourhood V of y n X such that cl(v) s copact and cl(v) U Then C = cl(v) s a copact set n Y contanng the neghbourhood V of y n Y Hence the proof of the theore s coplete Corollary 42 [2] : A space X s hoeoorphc to an open subset of a copact Hausdorff space f and only f X s locally copact Hausdorff A copactfcaton of a space X s a copact Hausdorff space Y contanng X such that X s dense n Y(That s, such that cl(x) = Y) Two copactfcaton Y and Y 2 are sad to be equvalent f there s a hoeoorphs h: Y Y 2 such that h(x) = x for every x X In order for X to have a copactfcaton, X ust be copletely regular, because a subset of a copact Hausdorff space s necessarly copletely regular Conversely, every copletely regular space has at least one copactfcaton One way of obtanng a copactfcaton of X s as follows: Suppose X s copletely regular space Choose an beddng h: X Z of X n a copact Hausdorff space Z Let X denote the subspace h (X) of Z, and let Y denote ts closure n Z Then Y s a copact Hausdorff space and c(x ) = Y ; Therefore, Y s copactfcaton of X Let Y be the space [,] Then Y s copactfcaton of X = (,) It s obtaned by adjonng one pont at each end of (,) Let X be a copletely regular space Let {f α } α J be the collecton of all bounded contnuous real valued functons on X, ndexed by soe ndex set J For each α J, choose a closed nterval I α n R contanng f α (X) To be defnte, choose I α = [glb f α (X), lub f α (X)] Then defne, h: X I by the rule h (x) = (f α α (x)) α J Then by Tychonoff theore s copact Snce X s I α copletely regular, the collecton {f α } separates ponts fro closed sets n X Therefore, by the Ibeddng theore h s an beddng The copactfcaton of X nduced by h s called the Stone-Cech copactfcaton of X t s denoted by β ( Χ) The property of the Stone-Cech copactfcaton s the followng: Theore 44 [2] :(Extenson property) be ts Let X be a copletely regular space and let β( Χ) Stone-Cech copactfcaton Then every bounded contnuous real-valued functon on X can be unquely extended to a contnuous real-valued functon on β ( Χ) Unqueness of the extenson s a consequence of the followng lea: Lea 42 [2] :(Unqueness of the extenson) Let A X; let f: A Z be a contnuous ap of A n to the Hausdorff space Z There s at ost one extenson of f to a contnuous functon g: c (A) Z Suppose that g, g : c (A) X are two dfferent extensons of f; Choose x so that g(x) g (x) Let U and U be dsjont neghbourhoods of g(x) and g (x), respectvely Agan choose a neghbourhood V of x so that g(v) U and g (V) U Now V ntersects A n soe pont y; then g(y) U and g (y) U But snce y A, we have g(y) = f(y) and g (y) = f(y) Ths contradcts the fact that U and U are dsjont Ths copletes the proof of the theore The followng theore characterzed the fact that Stone- Cech copactfcaton n a space s unque Theore 45 [2] : Let X be copletely regular Let Y and Y 2 be two copactfcaton of X havng the extenson property Thus there s a hoeoorphs Φ of Y onto Y 2 such that Φ (x)

7 Pure and Appled Matheatcs Journal 24; 3(5): 5-2 = x for each x X 5 σ-copact and Fnally Copact Spaces A topologcal space X s sad to be σ-copact f and only f X s the unon of a countable faly of ts copact subsets Theore 5 [7] : Every closed subset of a σ-copact space s σ-copact Let C be any closed subset of a σ-copact space X Then X s the unon of a countable faly {K n : n N } of ts copact subsets Hence C K n s a closed subset of the subspace K n of X for each n N Hence C K n s copact n C for each n N Consequently C s the unon of the countable faly {C K n : n N } of ts copact subsets Thus C s σ-copact Theore 52 [7] : Every separated locally copact space X wth a countable open base s σ-copact Let α be any countable open base for X Let x be any pont of X Then x has a copact closed neghbourhood K Consequently there exsts a eber M of α such that x M K We know that f B and C are any subsets of a topologcal space X and f B C then B C and we also know that a subset B of a topologcal space X s closed f and only f B = B Snce K s closed, hence fro the above theore we have M K Snce K s a separated copact subspace of X and we know that every copact subset of a separated space s closed Hence fro the above M s a copact subset of K and so of X Thus for each x n X there s a eber M of the countable open base α such that x M It follows that X s the unon of the countable faly { M : M α} of copact subsets Theore 53 [7] : If X s a σ-copact locally copact separated space then there exsts a sequence (B n : n N) of copact subsets of X wth the property that ( B : n N) covers X and for each n copact subset C of X there s an n n N such that C B n Theore 54 [7] : Let X be a locally copact separated space and let X * = X { } be a one-pont copactfcaton of X Then X s σ- copact f has a countable local base n X * A topologcal space X s sad to be fnally copact or a Lndeloff space f and only f every open cover of X has a countable subcover A subset B of X s sad to be fnally copact f and only f B s fnally copact as a subspace of X Theore 55 [7] : Every topologcal space wth a countable open base s fnally copact Let β be an open cover of a topologcal space X wth a countable open base γ Then for each x n X there s a eber B of β such that x B and we know that a subclass β of a topology for a set X s a base of f and only f for each U n and each x n U there s a eber V of β such that x V U Hence fro the above theore, there s a eber C x of V such that x C x B Consequently the subclass {C x : x X} of γ s a countable cover of X For each C x we can pck a eber B x of β such that C x B x Then {B x : x X} s a countable subcover of β Hence X s fnally copact Theore 56 [7] : Every σ-copact space s fnally copact Let X be a σ-copact space Then there s a countable faly {X n : n N } of copact subsets of X such that X = {X n : n N } Consder any open cover α of X For each n n N there s a fnte subclass α n of α coverng X n, because X n s copact Hence {α n : n N } s a countable subclass of α and covers X Hence X s fnally copact Theore 57 [7] : Every regular fnally copact space s noral Let A and B be any two dsjont closed subsets of a regular fnally copact space X Snce X s regular, there exsts an open set U x for each x n A such that x U x U x (X B) The class {U x : x A} covers A Therefore {U x : x A} {X A} s an open cover of X Snce X s fnally copact, there exts a countable subclass of ths cover coverng X and therefore A, and the closure of each eber of ths countable class s dsjont fro B By an analogous arguent there exsts a countable class of open subsets of X such that ths class covers B and the closures of each eber of whch s dsjont for A Hence A and B have dsjont neghbourhoods Thus X s noral 6 Concluson In ths study we show that copactness, lt pont copactness and sequentally copactness are equvalent n etrzable spaces We ntroduce t here as an nterestng applcaton of the Tychonoff theore We show that every copact space s locally copact but not conversely We also show that the product of fntely any copact spaces s a locally copact space We show that Stone-Cech copactfcaton n a space s unque We also show that every regular fnally copact space s noral Ths research work would gve soe rearkable results whch can be used to study the whole topologcal space by studyng a fnte nuber of open sets References [] Seyour Lpschutz, General Topology, McGraw-Hll Book Copany, Sngapore, 965 [2] Munkres, Jaes R, Topology, 2nd edton, Prentce Hall, 2

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