Question

8. Let T be the topology on N which consists of Ø and all subsets of N of the form An={n, n+1, n + 2,...} where n EN (i) Determine the closed subsets of (N, T)

Answer 1

8.7 Sot", Xal, T = $ ¢, An Inen} where An> {m, nti, m+2, ...} o since An et unen - Each A is Open in x . 2) X-An is closed in x for each non " Ba xr An= {1,2,... nah is closed set - en X MEN an> a B={1,2,...,n} imen in closed set in X . 0 0 - 37,24,44,85} = closure of $ 7,24, 47,853 ...,854 1 fixe T بن) Ã => Xe closure of a = NE A or x E A'= set of limit points of Al ... it for any mobil breighbourhood ul of an An (u-4x3) + 0 Now consider for any men ; Ism=85 Am ; men alsmees s. . Now the neighbourhoorhs of m en are A ,, A2,..., Am and each Ano An (Am-{m2) 70 a Ism< 85
1
and 85 E A so that Ā = {1, 2, 3, ... ,854 1 [ Note: 864 ñ as 86¢ A and AB = {86, 87, ... 4 is open neighbourhood of 86, in x and An Agn-{86} = [lly for othene nen & ny, 86 .; n& 0 Ã . Again closure of $3,6,9,12}={1,2,3,..., 12 h . l same as what we above. ; (11 since if D is deuse in n . Then Don, whese closure of D = Ō. Take any infinite set in n, is deuse in on, or If, x-Damaia finite set, - Then D is pense in n . Since for any non, the open subhd. An of n, is infinite and Ana&n, mith,...} intersects with every infinite sett
2
in N, Junce on each men, is limit point of D we have either . or mod" Exe ÃEX Either at A To => Ō N. Os (U-{x3) na # . . where uies neighbourhood to of . Bo that D is deuse in n, is X-D = N-D= pe is finite set . . 1.
3