TITLE: some confusion in Munkres topology?
QUESTION [0 upvotes]: AS  i was  reading the  munkre topology   as  im not able to understand the exact meaning  as i under line in red line  and  Diagram also  i didn't understands

PLiz help me  and pliz exaplain me in detailed with simple language  so that i can easily understands
Thanks in advance

REPLY [1 votes]: The set $\{\frac{1}{2}\} \times (\frac{1}{2},1]$ is open in the subspace topology from $\mathbb{R} \times \mathbb{R}$ (in the order topology) as we can write it as (illustrated in the left picture):
$$\{\frac{1}{2}\} \times (\frac{1}{2},1]  =((\frac{1}{2},\frac{1}{2}), (\frac{1}{2}, 2))_{\text{lex}} \cap (I \times I)$$
where we write the set as an intersection of an open interval in the lexicographic order on the plane (so by definition an open set in that order topology) with the subspace $I \times I$. It's clear by the definition of the lexicographic order that $((\frac{1}{2},\frac{1}{2}), (\frac{1}{2}, 2))_{\text{lex}} = \{\frac{1}{2}\} \times (\frac{1}{2}, 2)$.
On the other hand, if we look at $I \times I$ as an ordered space in its own right, with the same lexicographic order, we see that the set $O = \{\frac{1}{2}\} \times (\frac{1}{2},1]$ is not open in this order topology: the point $p = (\frac{1}{2},1) \in O$ is not an interior point of $O$, because if it were, there would have to be an open interval $((a,b), (c,d))$ (in the lexicographic order restricted to $I \times I$ (so $(a,b), (c,d) \in  I \times I$ and the interval is just all points of $I \times I$ that lie lexicographically in-between) such that $p \in ((a,b), (c,d)) \subseteq O$. (This holds as $p$ is not the maximum or minimum of $I \times I$ so it has a local base of open intervals). In the right hand picture the supposed $(a,b)$ is the point on the left (drawn as $(\frac{1}{2}, \frac{1}{2})$ which we could take if we'd like, but the point $(c,d)$ is on the right, so must obey $c > \frac{1}{2}$ (as $d$ cannot be bigger than $1$, the first coordinate has to be bigger), and then we can find loads of points in the interval $((a,b), (c,d))_{\text{lex}}$ with $x$-coordinate larger than $\frac{1}{2}$ as well, contradicting that $((a,b), (c,d)))_{\text{lex}} \subseteq O$.
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