Question

Let and define $LaTeX: f:\mathbb{R}\to Y$ by $LaTeX: f(x)= \begin{cases} 1&\text{ if }x<0\\ 2 &\text{ if }x=0\\ 3&\text{ if }x>0 \end{cases}$ . Describe the quotient topology on induced by by assuming that $LaTeX: \mathbb{R}$ has the usual topology.

Answer 1

Ams: Let Y={1, 2, 3} J: R y is defined as I enco 2 , x=0 3, ao Here IR is induced with usul topology Kit e be any subred of Y. For the quotient topology on y we need o subset Ty is open if ft (6) is open in R. 1. Ef e=q3 then fr (6) = (0,0) so. f (o) is open in R with perfect to usual tomology 1. Ef oz qo} then f (0) = 86] . so, t (o is closed in R with perfect - to usual topology. Ef 6 -43] then f (0) = (0,0) - so, ft(e) is open R with perfect to usual topology.

af then e f denote the quotient topology on y t f = {0, 1, 113, 93]{1,3}]