Properties of dictionary order topology

Properties of dictionary order topology

Choose the correct option

My attempt : According to Munkres the order topology on $\mathbb{R} \times \mathbb{R}$ has as basis the collection of all open interval of the forms ( $a\times b$ ,$ c \times d$) for $a < c$ and for $a =c$ and $b <d$

so according to the definition option c) will be the correct answer

Is its true ?

Any hints/solution will be appreciated

thanks u

1 Answer

The base for the topology is all sets of the form $[0\times 0, a \times b)$ and $(a \times b, 1\times 1]$ and all open intervals of the form $(a \times b, c \times d)$ where both endpoints lie in the square. (we have to treat the minimum and maximum of the set a bit differently).

I think (C) is indeed the correct option.