---- > [!definition] Definition. ([[Baire space]]) > A [[topological space]] $X$ is said to be a **Baire space** if the following condition holds: given any [[countably infinite|countable collection]] $\{ A_{n} \}$ of [[closed set|closed sets]] in $X$, each having empty [[topological interior|interior]], their union $\bigcup_{n}A_{n}$ also has empty [[topological interior|interior]] in $X$. > ^definition > [!equivalence] > - $X$ is a Baire space if and only if given any countable collection $\{ U_{n} \}$ of open sets in $X$, each of which is [[dense]] in $X$, their intersection $\bigcap_{n}^{}U_{n}$ is also [[dense]] in $X$. > > (This follows immediately from the straightforward fact that $B$ has empty interior in $X$ if and only if $\overline{X-B}=X$.)[^1] > > ^equivalence > [!equivalence] > A subset $A \subset X$ is called **meager**, or of the **first category**, in $X$ if it is contained in the union of a [[countably infinite|countable]] collection of [[closed set|closed sets]] in $X$ having empty [[topological interior|interior]]. It is called **non-meager**, or of the **second category**, in $X$ otherwise. > > Using this terminology, we can say the following: > - $X$ is a Baire space if and only if every nonempty open subset of $X$ is non-meager. [^1]: Assuming this fact (draw a picture) A given collection $\{ A_{n} \}$ of closed sets determines a collection $\{ U_{n} \}$ of open sets via $U_{n}=X-A_{n}$, and conversely. One has $\bigcup_{_{n}}A_{n}=X- \bigcap_{n}^{}U_{n}.$Suppose $X$ is a Baire space. Let $\{ U_{n} \}$ be a countable collection of dense open subsets of $X$. Then $\bigcap_{n}^{}U_{n}=X-\bigcup_{n}A_{n}$, [[closure|whence]] $\overline{\bigcap_{n}^{}U_{n}}= \overline{X - \bigcup_{n}A_{n}}$which equals $X$ by the initial remark. Converse is similarly easy. > [!basicnonexample] > $\mathbb{Q}$ is not a Baire space with the [[subspace topology]] inherited from $\mathbb{R}$. Indeed, each singleton $\{ q \} \in \mathbb{Q}$ is [[closed set|closed]] with manifestly empty interior, but the union of these singletons is $\mathbb{Q}$, and $\mathbb{Q}$ does not have empty interior ($\mathbb{Q} \cap U$ is open in $\mathbb{Q}$ for all $U$ open in $\mathbb{R}$). ^nonexample > [!basicexample] > $\mathbb{N} \subset \mathbb{R}$ is (somewhat vacuously) a Baire space, since every subset $U \subset \mathbb{N}$ is open (hence has nonempty interior if nonempty, for $\text{int }U=U$). More generally, every [[subspace topology|closed subspace]] of $\mathbb{R}$, being a [[complete]] [[metric space]], is a Baire space by the [[Baire category theorem]]. ^basic-example ---- #### ---- #### 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 > ```