Question

Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of Hausdorff. If not, show how the definition fails.

Answer 1

Detax and Y be two topological space with a purjee fine map Pixy, then Pis a guotient map and for any open set ucy Pi (u) is open and vice- - versa. I had a This map defines quotient topology wet, PiRR /IN (ul be the quotient map defined by x [x] = x + 1 Now take any two element welt X+ I ,YI ERIT [where x,y & I Now, x + 1 = {u} , y+I= {y} Take two rbd. uofa and vofy sit, unv=8 in IR. Then, P(u) and P (v) are two noods. of [x] and [y] in RI and, P (u) P (v) = 0 a (1) Now take x + I = {x} and I, the two elements of R/I. where x& I

- this is possible Let U and v two open set in R such that, [0, 1] cu and R EV and unv=0 [AS, IR is Normal, la Then, I EP (u) and [x] EP (v) Where, P(w) and P (v) are two open sets in R/I Where, P(u) n P (v) = 6 - (2) From (1) and (2) We have , R2 is Hausdorff. at an asset to I pri svoj cy} spa Itti In cat se ( W in (1) will Tim [ ] 6 mo [x] to taste Ibum &x - Its wat non T rowe ost who