Existence of infinite discrete family of open sets in a non compact topological space How can I prove if a Hausdorff topological space $X$ is not compact, then there exist a countably infinite discrete family of open sets in $X$.
 A: I don't think this is true.  Note that any Hausdorff space $X$ which has an infinite discrete family of (nonempty) open sets cannot be countably compact, and there are certainly examples of Hausdorff countably compact non-compact spaces (e.g., the ordinal space $\omega_1$).
For more detail, suppose $\{ U_n \}_{n \in \omega}$ is a discrete family of nonempty open subsets of $\omega_1$.  Without loss of generality, we may assume that each $U_n$ is of the form $( \alpha_n , \beta_n )$ for $\alpha_n < \beta_n$.  Letting $\gamma = \sup_{n < \omega} \beta_n < \omega_1$, note that each $U_n$ is actually a subset of the ordinal space $[0,\gamma]$, which is compact.  But this contradicts the assumed discreteness of the family in question (since no compact space can have such a family).
A: As Arthur Fischer points out, the result is false. What you can prove is that if $X$ is a non-compact Hausdorff space, there is an infinite family of pairwise disjoint open sets in $X$.
Let $I$ be the set of isolated points of $X$. If $I$ is infinite, then $\{\{x\}:x\in I\}$ is an infinite family of pairwise disjoint open sets, so assume that $I$ is finite. Replacing $X$ by $X\setminus I$ if necessary, we may assume that $I=\varnothing$. Fix distinct $x_0,x_1\in X$, and let $U_0$ and $V_1$ be disjoint open nbhds of $x_0$ and $x_1$, respectively. The point $x_1$ is not isolated, so there is a point $x_2\in V_1\setminus\{x_1\}$; let $U_1$ and $V_2$ be disjoint open sets such that $x_1\in U_1\subseteq V_1$ and $x_2\in V_2\subseteq V_1$. 
In general, given $x_n\in V_n$, choose $x_{n+1}\in V_n\setminus\{x_n\}$, and let $U_n$ and $V_{n+1}$ be open sets such that $x_n\in U_n\subseteq V_n$ and $x_{n+1}\in V_{n+1}\subseteq V_n$. It’s clear from the construction that if $m<n$, then $U_n\subseteq V_{m+1}\subseteq X\setminus U_m$, so $\{U_n:n\in\omega\}$ is the desired family of pairwise disjoint non-empty open sets.
Of course the family $\{U_n:n\in\omega\}$ is discrete in the subspace $\bigcup_{n\in\omega}U_n$, but in general that’s the most that you can guarantee.
