# inductive limit of compact Hausdorff spaces being realcompact

Suppose that $\{X_n\}_{n\in \mathbb{N}}$ is an increasing sequence of compact Hausdorff spaces and $X$ is their union, equipped with the finest topology under which all the inclusion maps are continuous. Is it well-known that $X$ is realcompact? Is there a reference or an easy argument for this fact? Thank you very much.

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After chapter 4, Rings of Continuous Functions' apparently assumes all spaces are completely regular. In particular the general claim that all Lindelof spaces are realcompact' appears incorrect. For example every non Hausdorff finite space is Lindelof, but no such space is realcompact. Given the hypothesis of the orginal question, can anyone clarify the following: 1) Must X be Hausdorff? 2) If X is Hausdorff, must X be regular? – Paul Fabel Oct 17 '12 at 15:58

Clearly your $X$ is Lindelöf, so it follows from Theorem 8.2 in Gillman & Jerison, Rings of Continuous Functions: Every Lindelöf space is realcompact.