Demostration of Taimanov's Extension Theorem

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SUMMARY

The Taimanov's Extension Theorem states that if A is dense in a T_1-space X, a continuous function f from X to a compact space Y has a continuous extension f^*: X → Y if and only if for every two disjoint closed sets F_1 and F_2, the preimages f^{-1}[F_1] and f^{-1}[F_2] have disjoint closures in X. The demonstration of this theorem can be found in Ryszard Engelking's "General Topology" (Theorem 3.2.1) and is also available in the original Russian document linked in the discussion.

PREREQUISITES
  • Understanding of T_1-spaces in topology
  • Familiarity with continuous functions and compact spaces
  • Knowledge of closed sets and their properties in topological spaces
  • Basic proficiency in mathematical Russian for accessing original documents
NEXT STEPS
  • Study Ryszard Engelking's "General Topology" for a comprehensive understanding of the theorem
  • Research the properties of T_1-spaces and their implications in topology
  • Explore the concept of continuous extensions in topological spaces
  • Review disjoint closed sets and their closures in the context of topology
USEFUL FOR

Mathematicians, particularly those specializing in topology, graduate students studying advanced topology concepts, and anyone interested in the applications of Taimanov's Extension Theorem.

heras1985
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I need the demostration of the Taimanov's extension theorem:
This theorem said:
Let [tex]A[/tex] be dense in the [tex]T_1[/tex]-space [tex]X[/tex]. Then in order that a continuos function [tex]f[/tex] from [tex]X[/tex] into a compact space [tex]Y[/tex] have a continuous extension [tex]f^*:X\rightarrow Y[/tex] if and only if that for each two disjoint closed sets [tex]F_1[/tex] and [tex]F_2[/tex], [tex]f^{-1}[F_1][/tex] and [tex]f^{-1}[F_2][/tex] have disjoint closures in [tex]X[/tex].
Where can I find the demostration of this theorem?
Thanks
 
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