I'm taking an introductory course in topology and I came across the following problem.
Decide if $X=[0,1) \times [0,1]$ with the topology from the lexicographical order is connected or not.
When I started thinking about the problem I didn't have much intuition about this topological space so I tried to prove that see if it was disconnected. For that I had seen before that the space $[0,1) \times \mathbb N$ was disconnected by the open sets $U= \{0\} \times \mathbb N$ and $V=(0,1) \times \mathbb N$. I tried to replicate that same idea. I realized that the set $W \subset X$ given by $\{0\} \times [0,1]$ wasn't an open set so that I couldn't make the same separation as before. This gave me the idea that it was connected but to prove that something is connected it tends to be a little more complicated. I tried to find a suitable homeomorphic topological space but couldn't quite get an idea of which one. I would gladly accept any hints.
Best Answer
$X = [0,1) \times [0,1]$ obeys the conditions that Munkres' states before theorem 24.1 of being a "linear continuum". One can check this essentially the same way as Munkres does (in Example 1 following 24.3) for the whole $[0,1]\times[0,1]$ (our $X$ is the full square minus the final interval $\{1\} \times [0,1]$, which leaves it connected). So $X$ is connected, as it is an order-convex subset of the linear continuum $[0,1]\times [0,1]$. (This is an alternate proof besides checking the linear continuum properties directly. Note that $X$ equals the set of all $(x,y)$ that are $< (1,0)$, e.g.)
$Y=[0,1] \times [0,1)$ is not connected because we're removing a lot of loose interior points, as it were (all points of the form $(x,1)$). $\{0\} \times [0,1)$ is a closed-and-open subset of $Y$, e.g. (all points below $(0,1)$ and above $(0,1)$ from the full square).
Be careful, the lexicographical order needs careful consideration...