presentations for bs students, Summaries of Topology

Download presentations for bs students and more Summaries Topology in PDF only on Docsity! aca aaa Let (X, T) be a topological space and let A < X, then the inter- section of all closed set which are superset of A is called the closure of A and is denoted by A. Example : Let X = {a, b, c, d, e} T = {, X, {a}, fa, b}, {a, c, d}, {a, b, e}, {a, b, c, d}} Then find closure of {c, e} Solution : The closed subset of X are , X, {b, c, d, e}, {c, d, e}, {b, e}, {c, d}, {e} then A= {c, e}={b, 6, d,e} 0 fc, d,e} aX = {c, d, e} 5  Let (X, T) be topological space and let A, B < X, then 1. Ais said to be dense in Biff BCA. 2. Ais everywhere dense in X iff A = X. 3. Ais no-where dense in X iff (A)° = 6. 4. Ais dense in itself iff A < D(A). Let X = {a, b, c, d, e} and = T = {, X, {a}, {b, c}, fa, b, c}} | then check the set A = {a, d} is dense in B = {a, b, c, d}. Solution : Closed set are , X, {b, c, d, e}, {a, d, e}, {d, e} So, A= {a, d, e} but BcA => Ais not dense in B. Let X = (a, b, c} and T = (6. X. {b), {a,b}, then closure of A= {b, c} Q {b. ¢} g {a, b. c} @ @) Q «© Let R be the set of all real numbers. Let U be the family of subsets of R consisting of # and all non-empty subset G of R having the property that to each x € G, 4 an open interval |, such that x < |, < G, then U is topology on R, called the usual topology i.e. U = {(a, b), (c, d), (a, b) u (ec, d), «0... } OR Let U consist of » and all those subset G of R which are neighbourhood of each x € G. The topological space (R, U) is called usual topological space. _ — o- ow iG) Q. LOWER LIMIT TOPOLOGY FORR Let R be the set of all real numbers. Let S be the family of all subset of R consisting of > and all non-empty subset G of having the property that to each x € G J a right half open interval [a, b[ s.t. x € [a, b) c G, then S is called lower limit topology for R. - ia , L T T a x b The topological space (R, S) is called lower limit topological space. Note : (i) Every open interval containinS. 9 ~——*+3-+—+— (ii) Interval of type [a, b) contained in S. —1 ae (iii) (a, b] ¢ S because b « (a, b] but # any right half open interval [b, p) > ___ s.t.b efb,p) < (a, b]  Let R be the set of all real numbers. Let H be the family of all subset of R consisting of and all non-empty subset G of having the property that to each x <« G 4 a left half open interval (a, b] s.t. x € (a, b] c G, then H is called upper limit topology for R. The topological space (R, H) is called upper limit topological space. GcR —_ ro) | T a x b  Show that the every closed interval [a, b] is closed in the usual topology for R. Solution : G = [a, b]’ = (-», a) U (b, ~) G is neighbourhood of each point x <« G. So, (-%, a)U (b, 0) e U => (-», a) U(b, ») is open set => [a, b] is U-closed set.