Question

15.1. Which of the following spaces are Hausdorff: (1) a discrete space; an indiscrete space; (2) (3) the arrow; (4) RT (5) ? for a minute, we a oven

Hello, I am trying to solve problem 15.1 which is shown in the first image. I attached extra images that explain what an arrow mean in the question and what the 4 dot symbol mean. The RT1 is the separation axiom 1. I also attached what is meant by a discrete and indiscrete space. The last image give the answer from the book, but I need an explanation. this problem is from "Elementary topology problem textbook". Please have clear hand writing and I appreciate your time.

Answer 1

& in a topological space (x,-2), au call elements of the topology or as opin sets. (15.) : 1) Hausdorff Space:- for every pair at point xey ex, there exist two open sets u į v elements of a such that neur you and unv=0. then . ( x2) is Hausdorf. Since in discrete space, every subset of 8 is an element of a lie., & every subset ab x is an open set. so, each singhton's are also open sets. so, for each ni gfx, U=204 Iv=ays gives the facility to call (X12) (a discrite spac) to as Haus douff space. o open sets in an indiscrete spacey are Just t itulf and d. So, for any HigEX , we cannot find in sets ut v such that, KEUR YEV A4 Unveď so Inducrete space is nat Hausdorff

3) X= [0,00) (Arrows ) and & corists at elements of type (a, as), as, O. you so that Carco), as, o are the only open sets of your besides fo. And each open sets interruts another ow. for any a, b>0. • Ca, a) ( 5,a) - (5) + 6 7 8 = ca,co) if asb So, for any legfx, un con hat - get such open sets u d v . siti unu = $. thence Arrow is nat Hausdorff. (4) Ret RTA An Rt, space is a topological spacet, in which for each pair x, yox, there is an open ut u r an open dit v such that x tu e gfu and x&v 4 4¢U our claim is, Rtg is Not Hausdorff in general. counter example: consider the RT, space X-1,2,34 tal21,24, {1,34, 82,345 for it i t 2, U = 21,34; V=22,35 TEU, 26 V 1e v260

for 283, Us P124, V-2134 . 260, 36V, 24V, 3¢ u 4 for the pair, 1, 43 U=2424, v=22,34 So, (x, 2) is Rt, space. but it is wat Hausdorff sinu. ",268 A Ang two open-eets containing 1 risputively interrut each other. X= r = 2 a, b, cids 0,8, 244, 234, Rarch, la, bich, kable 4 for are . the pair, a c EX outy candidates for veu Us 494, va 2016 but unv=hah Ulag, Valabics but unu=Lah Utact , V .6, . unv= 204 You can see that So, W is each such utu internut not Hausdorff