Analytic space | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-03-29

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Analytic_space


00:00:19 1 Definition
00:00:59 2 Basic results
NaN:NaN:NaN 3 Smoothness
NaN:NaN:NaN 4 Coherent sheaves
NaN:NaN:NaN 5 Generalizations
NaN:NaN:NaN 6 See also
NaN:NaN:NaN 7 References
NaN:NaN:NaN 1.  The circle is an analytic subset of the analytic space R2.  But its projection onto the x-axis is the closed interval [−1, 1], which is not an analytic set.  Therefore the image of an analytic set under an analytic map is not necessarily an analytic set.  This can be avoided by working with subanalytic sets, which are much less rigid than analytic sets but which are not defined over arbitrary fields.  The corresponding generalization of an analytic space is a subanalytic space.  (However, under mild point-set topology hypotheses, it turns out that subanalytic spaces are essentially equivalent to subanalytic sets.)
NaN:NaN:NaN See also



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SUMMARY
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An analytic space is a generalization of an analytic manifold that allows singularities.  An analytic space is a space that is locally the same as an analytic variety.  They are prominent in the study of several complex variables, but they also appear in other contexts.