Microsoft Word - Sitnshu S. Chaudhary _18-20_.doc Shitanshu Shekhar Choudhary and Raju Ram Thapa / BIBECHANA 7 (2011) 18-20 : BMHSS 18 BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (online) Journal homepage: http://nepjol.info/index.php/BIBICHANA Locally compact space and continuity Shitanshu Shekhar Choudhary , Raju Ram Thapa ∗∗∗∗ P.G. Campus, Biratnagar, Tribhuvan University Article History: Received 27 September 2010; Revised 1 November 2010; Accepted 7 November 2010 Abstract Topological spaces for being T0, T1, T2 and regular space have been discussed. The conditions for a topological space to be locally compact have also been studied. We have found that a continuous function preserves locally compactness. Keywords: Topological spaces; Compactness; Regular space 1. Introduction It has been seen that any product of compact spaces is compact. It has been also seen that most of the spaces turn out to be closed subspaces of products of compact spaces and such spaces are necessarily compact. The n-dimensional Euclidean space R n is the most important type of topological space which has a great importance in Modern analysis. A topological space is locally compact if each of it's point has a neighborhood with compact closure. As a result, R n is locally compact for any open sphere centred on any point, is the neighborhood of the point whose closure, being a closed and bounded subspace of R n is compact. It's application is in the field of geometry and Analysis. 2. Definitions 2.1 T0- space A topological space (X, J) is called T0 space or Kolmogorff space iff given any pair of points xy,x ∈ (distinct) there exists an open set containing one of them but not the other. 2.2 T1- space A topological space (X, J) is called T1- space or Frechet space ∗ Corresponding author: Raju Ram Thapa, Dept. of Mathematics, Post Graduate Campus, Biratnagar, Tribhuvan University, Email: thaparajuram@yahoo.com Shitanshu Shekhar Choudhary and Raju Ram Thapa / BIBECHANA 7 (2011) 18-20 : BMHSS 19 iff given any pair of points Xy,x ∈ there exist two open set, one containing x not y & other containing y not x. 2.3 T2- space A topological space (X, J) is said to be T2-space or Hausdroff space iff given any pair of points Xy,x ∈ there exits two disjoint open sets one containing x and other y. Regular space : A topological space X is called regular if for each closed subset F of X and Xx ∈ such that Fx ∉ , there exist disjoint open sets G and H such GF ⊆ and Hx ∈ . 2.4 T3 - space A regular T1-space is called T3-space. 2.5 Locally compact space A top. space (X, J) is said to locally compact if given Xx ∈ and any nbd. U of x, there is a compact set A such that .UAAx 0 ⊂⊂∈ 0 A : It is union of all open sets contained in A called interior of A, obviously AA 0 ⊂ . 3. Formalism Proposition: Let (X, J) be a T2-space then X is locally compact iff given Xx ∈ , there is a compact set A such that 0 Ax ∈ . Proposition: Any compact T2-space is locally compact. Proposition: Any locally compact T2-space X is T3. Proposition: If a space X is T2 and locally compact then every open and closed subspace is also T2 space and locally compact. Proposition: A substance Y of a locally compact T2-space X is locally compact iff it is the intersection of an open set and a closed set. Proposition: Let f be a continuous and open function from one topological space (X, J) to another topological space (Y, J') then X is locally compact Y⇒ is locally compact. Proof: Let (X, J) and (Y, J') be two topological spaces and f be a continuous and open mapping from X onto Y. Let X is locally compact. To show that Y is also locally compact, let Yy ∈ and N be the neighborhood of y then )x(fy = for some Xx ∈ . Since f is continuous so there exists a neighborhood v of x such that f(v) ⊂ N. Since X is locally compact so there exists a compact set B such that 0 Bx∈ NB ⊂⊂ . Then .N)B(f)B(fy)x(f 0 ⊂⊂⊂= But )B(f 0 is open, being f is open mapping and also compactness is invariant under continuous mapping so f(B) is compact. Thus N)B(f)B(fx 0 ⊂⊂∈ . Which shows that Y is locally compact. Shitanshu Shekhar Choudhary and Raju Ram Thapa / BIBECHANA 7 (2011) 18-20 : BMHSS 20 Proposition: Let Iii }J,x{ ∈ be a countable families of non-empty spaces and iX∏ be the product spaces. Then iX∏ is locally compact iff each component spaces is locally compact and all of the component spaces except atmost finitely many are compact. Proof: Let ip be the projection mapping i Ii ii XX:p →∏ ∈ which is continuous onto and open so each iX is locally compact. Let A be any compact subset of iX∏ such that some point y of ∏ ∈Ii iX is in 0 A . Then there is a basis neighborhood ∏ ∈Ii iV of y such that ii XV = for all but at most finitely many i and ∏ ∈ ⊂ Ii 0 i AV . Thus ii X)A(p = for all but at most finitely many i, since ip is continuous and A is compact so iX is compact for all but at most finitely many i. Now we assume that each iX is locally compact and all but finitely many of iX are compact. Let ∏ ∈ ∈ Ii iXY,X and let iY be the i th co-ordinate of Y. If U is any neighborhood of y then U contains a basis neighborhood of Y of the form ∏ ∈Ii iV , where iV is open in iX for each i and ii XV = for all Ii ∈ , except for at most finitely many say n21 i.,.........i,i . Since each iX is locally compact so for each Ii ∈ there is a compact subset iA of iX . Such that ii 0 ii VAAY ⊂⊂∈ . There are at most finitely many more Ii ∈ , other than n21 i.,.........i,i say m2n1n i,.........i,i ++ such that m2n1n Xi.........,,Xi,Xi ++ are not compact. For any i not in n21 i.,.........i,i , m2n1n i,.........i,i ++ we may let ii XA = then ∏∏ ∏∏ ∈∈ ∈∈ ⊂⊂⊂∈ Ii i Ii Ii i 0 i Ii i 0 VA)A(Ay . But ∏ ∈Ii iA is product of compact sets and is therefore compact. Hence ∏ ∈Ii iX is locally compact. 4. Conclusion Locally compactness is not invariant under continuous mapping but in under certain assumptions which are openness, a continuous function preserves local compactness. References [1] G. A. Quarw, Point countable open covering in countable compact space; Academic Press, New York (1967). [2] R. Arers, Remark on concept of compactness; Port Maths (1950). [3] N. Dykes, Pac.J. Maths(1970). [4] M. Henrikse and J.R., Isbell Duke Maths J. (1958). [5] H. Wickle & J.M. Worrel, Proc. Aurer, Maths society (1976).