---- > [!theorem] Theorem. ([[Baire category theorem]]) > If a [[topological space]] $X$ is a [[compact]] [[Hausdorff space]] or a [[complete]] [[metric space]], then $X$ is a [[Baire space]]. ^theorem > [!proposition] Corollary. > > Let $X$ be a nonempty [[complete]] [[metric space]]. Then: >1. $X$ is not the [[countably infinite|countable]] union of [[closed set|closed subsets]] with empty [[topological interior|interior]] (since $\text{int }X=X \neq \emptyset$). > 2. The [[countably infinite|countable]] intersection of [[dense]] open subsets of $X$ is nonempty (since $\emptyset$ is not [[dense]] in $X$) > [!proof]- Proof. ([[Baire category theorem]]) > Black-box for now, though will definitely return eventually. ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` Some different options: - probability until lunch - PDE exercise from lunch-dinner - Lie groups lecture after dinner