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generic (Definition)

A property that holds for all $ x$ in some residual subset of a Baire space $ X$ is said to be generic in $ X$, or to hold generically in $ X$. In the study of generic properties, it is common to state “generically, $ P(x)$”, where $ P(x)$ is some proposition about $ x\in X$. The useful fact about generic properties is that, given countably many generic properties $ P_n$, all of them hold simultaneously in a residual set, i.e. we have that, generically, $ P_n(x)$ holds for each $ n$.



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Also defines:  generically
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Cross-references: proposition, Baire space, subset, residual, property
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This is version 7 of generic, born on 2003-06-11, modified 2003-07-25.
Object id is 4341, canonical name is Generic.
Accessed 3706 times total.

Classification:
AMS MSC54E52 (General topology :: Spaces with richer structures :: Baire category, Baire spaces)

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