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| If $X$ is a Baire space, we say that a subset $S$ of $X$ is \emph{generic} (in $X$) if $X-S$ is meagre, or equivalently, if $S$ contains a countable intersection of open and dense sets. From the Baire property, we know that a countable intersection of generic sets is generic as well. |
If $X$ is a Baire space, we say that a subset $S$ of $X$ is \emph{generic} (in $X$) if $X-S$ is meagre, or equivalently, if $S$ contains a countable intersection of open and dense sets. From the Baire property, we know that a countable intersection of generic sets is generic as well. |
| \textbf{Remark} If a property holds for a generic subset of $X$, it is said that the property is generic in $X$, or that it \emph{holds generically} in $X$. In the study of generic properties, it is common to state ``generically, $P(x)$'', where $P$ is some proposition about $x\in X$. The useful fact about generic properties is that, if $P$ and $Q$ are two generic properties, then the property ``P and Q'' also holds generically. |
\textbf{Remark} If a property holds for a generic subset of $X$, it is said that the property is generic in $X$, or that it \emph{holds generically} in $X$. In the study of generic properties, it is common to state ``generically, $P(x)$'', where $P$ is some proposition about $x\in X$. The useful fact about generic properties is that, if $P$ and $Q$ are two generic properties, then the property ``P and Q'' also holds generically. |
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