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Let $X=Z_+×Z_2, Y=Z_2×Z_+$ with the dictionary order topologies, are $X$ and $Y$,$T_1$,$T_2$ and $T_3$?

$Z_+=\{1,2,3,...\}$

$Z_2 =\{0,1\}$

The open set on the dictionary ordering will be $I=\{x×y∈\mathbb{R×R}:a×b<x×y<c×d\}$

Now, how to deal with separation axioms $T_1$,$T_2$ and $T_3$?

I am a fresh masters student in pure mathematics. After several days, I have an exam in the Topology course while I was trying to solve of many questions I faced this question and I could not deal with it well, especially since the teaching of this semester is electronic. I want your help.

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Every space whose topology is determined in this way by a linear order is $T_1$ and hereditarily normal, so it is $T_1$, $T_2$, and $T_3$. The first two are very easy to prove.

Let $\langle X,\le\rangle$ be a linearly ordered set with the order topology: this is the topology generated by the subbase of open rays, the sets of the form $(\leftarrow,x)$ and $(x,\to)$. If it has no first or last element, it has a base consisting of the open intervals $(x,y)$ for $x,y\in X$ with $x<y$. If it has a first element $a$, one has to add the intervals $[a,x)$ for $a<x$ to the base, and if it has a last element $b$, one has to add the intervals $(x,b]$ for $x<b$ to the base.

Now let $x$ and $y$ be distinct points of $X$. Without loss of generality we may assume that $x<y$. Note that if we prove that $X$ is $T_2$, we can automatically conclude that $X$ is $T_1$. There are two cases; I’ll point them out but leave some of the work to you.

  • If there is no point $z$ such that $x<z<y$, i.e., if $(x,y)=\varnothing$, let $U=(\leftarrow,y)$ and $V=(x,\to)$. Verify that $U$ and $V$ are disjoint open nbhds of $x$ and $y$, respectively.
  • If there is a point $z\in(x,y)$, so that $x<z<y$, use $z$ to define disjoint open rays containing $x$ and $y$.

This is overkill for your problem, but it’s easy and worth knowing. A proof that spaces with linear order topologies are always $T_3$ is significantly harder, so for that part of the problem you should concentrate on your specific spaces. Show that one of them has the discrete topology, and the other has only one non-isolated point (i.e., one limit point). Once you’ve done that, you should have little trouble verifying that they are $T_3$, since you’ll know exactly what the closed sets are.

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