As the title says, I need to prove thath if $X$ is an infinite $T_3$ (here $T_3$ is $T_1$ + regularity) topological space then there exist $\mathcal{F}=\{U_n\mid n\in\mathbb{N} \}$ such that for all $n\in\mathbb{N}$, the set $U_n$ is open and if $n\neq m$ then $U_n\cap U_n=\emptyset$.
My attempt:
First, take two different points $x_1,x_2\in X$. By Hausdorfness of $X$, there exist $V_1, V_2$ a disjoint open sets such that $x_1\in V_1$ and $x_2\in V_2$. Take $x_3\in X\setminus\{x_1,x_2 \}$. Then, by the regularity of $X$, there exist $V_3$ and $V_4$ a disjoint open sets such that $x_3\in V_3$ and $\{x_1,x_2 \}\subseteq V_4$. Then take $U_1=V_4\cap V_1$, $U_2=V_4\cap V_2$ and $U_3=V_3$. Therefore $x_1\in U_1$, $x_2\in U_2$ and $x_3\in U_3$ and moreover, $U_1\cap U_2=\emptyset$, $U_1\cap U_3=\emptyset$ and $U_2\cap U_3=\emptyset$ and all of them are open sets. This step is like the basis of the induction.
Now, suppose that we have constructed $U_1,U_2,\dots,U_n$ a family of mutually disjoint non-empty open sets. Following the later construction, we can take $x_i\in U_i$ for $i\in\{1,\dots,n \}$. For $x_{n+1}\in X\setminus\{x_1,\dots,x_n \}$, by regularity, there exist $W_1$ and $W_2$ disjoint open sets such that $x_{n+1}\in W_1$ and $\{x_1,\dots,x_n \}\subseteq W_2$. But from here I'm stuck. What can I do? Any suggestion? Thanks.
Best Answer
By a classic theorem due to Ginsberg and Sand (which is not very well-known), but was proved here, if $X$ is any infinite topological space then $X$ contains a subspace homeomorphic to one of the following five spaces:
If $X$ is Hausdorff (or "better"), it cannot contain spaces 1-4, as these are not Hausdorff, so it has a countable discrete subspace, which implies the existence of the required $U_n$ to show every $\{n\}$ is open in the subspace. (with some minor modifications, we can also ensure that the $U_n$ are also disjoint on $X$.)