Show that the dictionary order topology on the set R × R is the same as the product topology ℝ_d × ℝ where ℝ_d denotes ℝ in the discrete topology. Compare this topology with the standard topology on ℝ^2.

1) Define the well-ordering property

Consider X = {a1, a2, ..., an}Can a topology of X with 3 open and one with 4 open be equivalent?

Consider the set A={a,b,c,d,e,f}. Define a partial ordering on A by setting x<=y if (x,y) belongs to R={(a,a)(b,b)(c,c)(d,d)(e,e)(f,f)(a,d)(a,f)(b,c)(b,d)(e,b)(e,c)(e,d)(f,d)} a)Find all x belongs to A that satisfy x<d b)find all x belongs to A that are incompsctable with c