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at Sis coarser (i, arger ) than 3!, , 9 cS 3, we say, ren we say that 4, , np, and 33 are all topologies for X and 3y is finer en Sh x :, Then topologies Sy and $3 are not comparable since 3, ¢ 33 and 93 ¢ 2S. si, topologies 35 and 33 are not comparable., , Example 14: Consider the usual topology U for Rand the lower limit topology S for R and show that, S is finer than OF, , Solution: We have to show that every U-open set is S-open. Let G € U. To show that, G eS. Let p be any point of G. Since G is U-open, there exists an open interval Ja, b[ with p, as mid point such that p €]a,b[cG,, , Now P €)4b[>[pbl c]ab[, , Thus to each p ¢ G3 half-open interval [p, b[ such that, pelpb[cG, , Hence G €S. Therefore U cS. This is, S in finer that U., , Intersection and Union of Topologies [Rohilkhand 2004], , Let X = {a,b,c}. Consider two topologies 3, and 3, for X defined as follows:, 3, ={G, {a}, X}, By = {, {b}, X}, Then 3 U 8p ={O, {a}, {b}, X}, , which is not a topology for X., , Thus the union of topologies is not necessarily a topology on X. However, the intersection, of any collection of topologies is a topology for X as our next theorem shows., , *“Theorem 1: Let {3, :d € A} where A is an arbitrary set, be a collection of topology for X. Then, , the intersection 0 {3,:A € Ajis also a topology for X., [Meerut 2002, 2006 (B.P.); Agra 2006; Bilaspur 2006, 2008;, C.S.I.R. (Net) 2005; Rohtak 2003, 2004; Garhwal 2009], , Proof: Here {3, che A} is a collection of topologies on X. We have to show that, {3 :he A}fisalsoa topology on X. If A = @,thenn {3, :’ € @} =P (X). [see *Remark, in (1.6.4), ch. 1]., , Thus in this case the intersection of topologies is the discrete topology., , Now let A #@., , (T1]: Since 3, isa topology VA € A, it follows that @, X € 3, ¥AeA., , But De VAEASPen{S, :r’ cA}, and XeS, VAEASX EN{S, :A EA}