equivalence of forcing notions

Let $P$ and $Q$ be two forcing notions such that given any generic subset $G$ of $P$ there is a generic subset $H$ of $Q$ with $\mathfrak{M}[G]=\mathfrak{M}[H]$ and vice-versa. Then $P$ and $Q$ are equivalent.

Since if $G\in\mathfrak{M}[H]$ , $\tau[G]\in\mathfrak{M}$ for any $P$ -name $\tau$ , it follows that if $G\in\mathfrak{M}[H]$ and $H\in\mathfrak{M}[G]$ then $\mathfrak{M}[G]=\mathfrak{M}[H]$ .

Title	equivalence of forcing notions
Canonical name	EquivalenceOfForcingNotions
Date of creation	2013-03-22 12:54:24
Last modified on	2013-03-22 12:54:24
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	5
Author	Henry (455)
Entry type	Definition
Classification	msc 03E35
Classification	msc 03E40
Synonym	equivalent
Related topic	Forcing
Related topic	ProofThatForcingNotionsAreEquivalentToTheirComposition