Michael selection theorem

In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a branch of mathematics, Michael selection theorem is a <a href="/facts/Selection_theorem/gbmNwG5L">selection theorem</a> named after <a href="/facts/Ernest_Michael/5MAnjj3J">Ernest Michael</a>. In its most popular form, it states the following:

Michael Selection Theorem—Let X be a <a href="/facts/Paracompact/rjTrvF7j">paracompact</a> space and Y be a separable <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>. 
Let 
 
 
 
 F
 :
 X
 →
 Y
 
 
 {\displaystyle F\colon X\to Y}
 
 be a <a href="/facts/Hemicontinuity/HTtnNmuN">lower hemicontinuous</a> <a href="/facts/Set-valued_function/WH5ua6BL">set-valued function</a> with nonempty <a href="/facts/Convex_set/vdAuJRJl">convex</a> <a href="/facts/Closed_set/upYnRTUj">closed</a> values. 
Then there exists a <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a> <a href="/facts/Choice_function/AutN9YcF">selection</a> 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 of F.

<a href="/facts/Converse_(logic)/VhlUrxxx">Conversely</a>, if any lower semicontinuous multimap from topological space X to a Banach space, with nonempty convex closed values, admits a continuous <a href="/facts/Choice_function/AutN9YcF">selection</a>, then X is paracompact. This provides another characterization for <a href="/facts/Paracompact_space/rjTrvF7j">paracompactness</a>.

Michael selection theorem open-in-new

Michael selection theorem