My proof: Let $X$ be an space with the order topology, $x \in X$ and $F$ a closed set that does not contain $x$. Then, the set $X-F$ is an open set that contains $x$, hence there is an open set (basic) $(a,b)$ such that $x \in (a,b)\subseteq X-F$. Then $(a,b)$ and $(-\infty,a) \cup (b,\infty)$ […]

From Munkres, p.90 example 3: Let $I=[0,1]$. The dictionary order on $I\times I$ is just the restriction to $I\times I$ of the dictionary order on the plane $\mathbb{R}\times\mathbb{R}$. However, the dictionary order topology on $I\times I$ is not the same as the subspace topology on $I\times I$ obtained from the dictionary topology on $\mathbb{R}\times\mathbb{R}$! For […]

Let $Y$ be an ordered set in the order topology with $f,g:X\rightarrow Y$ continuous. Show that the set $A = \{x:f(x)\leq g(x)\}$ is closed in $X$. I am completely stumped on this problem. As far as I can tell I’ve either got the task of proving $A$ gathers its limit points, or showing that there […]

I’m trying to prove (or disprove) that every linear ordered set $(X, <_X)$ is normal in its order topology. I was able to prove $(X,<_X)$ is hausdorff, simply by taking two open intervals with $\pm\infty $ for every $x,y\in X$ with no common points, but when it comes to proving $(X,<_X)$ is normal, I’m not […]

Greets This is a problem I wanted to solve for a long time, and finally did some days ago. So I want to ask people here at MSE to show as many different answers to this problem as possible. I will offer a Bounty in two days, depending on the interest in the problem, and […]

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