12 research outputs found

    Remarks to the problem of defining a topology by its homeomorphism group

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    Zur Topologisierung metrischer Räume. III

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    Some remarks on isomorphisms of function algebras

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    Sia B(X)B(X) l'algebra reale di tutte le funzioni limitate a valori reali f:XRf:X\rightarrow\mathfrak{R} sull'insieme XX\neq\oslash . In questa nota mostriamo che ogni isomorfismo Φ\Phi fra le algebre B(X),B(Y)B(X),B(Y) è indotto da una biiezione φ:XY\varphi:X\rightarrow Y Φ(f)(t):=f[φ1(t)]. \Phi(f)(t):=f\left[\varphi^{-1(t)}\right]. Infine proviamo un teorema riguardante la rappresentazione di isomorfismi di certi sottomonoidi moltiplicativi di B(X)B(X) e accenniamo ad un'applicazione alla topologia generale.\ensuremath{B(X)} be the real algebra of all bounded real-valued functions f:XRf:X\rightarrow\mathfrak{R} on the set XX\neq\oslash. We prove in this note, that every algebra isomorphism Φ:B(X)b(Y)\Phi:B(X)\rightarrow b(Y) is induced by a bijection φ:XY\varphi:X\rightarrow Y Φ(f)(t):=f[φ1(t)]. \Phi(f)(t):=f\left[\varphi^{-1(t)}\right]. Finally we prove a theorem concerning the representation of isomorphisms of certain multiplicative submonoids of B(X)B(X) and mention an application in general topology

    Advanced research on acteoside for chemistry and bioactivities

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    Specification of topological spaces by algebraic systems of continuous functions

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