Editing Open mapping theorem

'''Open mapping theorem''' may refer to:

* [[Open mapping theorem (functional analysis)]] (also known as the Banach–Schauder theorem), states that a surjective continuous linear transformation of a Banach space ''X'' onto a Banach space ''Y'' is an open mapping
* [[Open mapping theorem (complex analysis)]], states that a non-constant holomorphic function on a connected open set in the complex plane is an open mapping
* Open mapping theorem ([[topological groups]]), states that a [[surjective]] continuous [[homomorphism]] of a locally compact [[Hausdorff space|Hausdorff]] group ''G'' onto a locally compact Hausdorff group ''H'' is an open mapping if ''G'' is ''σ''-compact. Like the open mapping theorem in [[functional analysis]], the proof in the setting of topological groups uses the [[Baire category theorem]].

== See also ==
* In [[calculus]], part of the [[inverse function theorem]] which states that a continuously [[derivative|differentiable]] function between [[Euclidean spaces]] whose [[Jacobian matrix and determinant|derivative matrix]] is invertible at a point is an open mapping in a neighborhood of the point.  More generally, if a mapping ''F''&nbsp;:&nbsp;''U''&nbsp;→&nbsp;'''R'''<sup>''m''</sup> from an [[open set]] ''U''&nbsp;&sub;&nbsp;'''R'''<sup>''n''</sup> to '''R'''<sup>''m''</sup> is such that the [[Jacobian]] derivative ''dF''(''x'') is [[surjective]] at every point ''x''&nbsp;∈&nbsp;''U'', then ''F'' is an open mapping.
* The [[invariance of domain]] theorem shows that certain mappings between subsets of '''R'''<sup>''n''</sup> are open.

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