On the Tietze extension theorem, if instead of a continuous function "f" we use a lower semi continuous function on a closed subspace of a metric space, is the theorem correct? I mean, can we extend every lower semi continuous function from a closed subspace of a metric space to the space?
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3$\begingroup$ Yes, at least for bounded functions. A function is lower-semicontinuous iff it is the pointwise supremum of a family of continuous functions. Just extend each continuous function in the family and take the pointwise supremum. $\endgroup$– Michael GreineckerMar 18, 2018 at 13:38
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3$\begingroup$ @MichaelGreinecker: If the function $f$ is bounded above, say $f \le M$, then one can simply extend the function by letting it equal $M$ on the complement of our closed set $E$. If you allow the extended reals, you could also set $f = +\infty$ on $E^c$. $\endgroup$– Nate EldredgeMar 18, 2018 at 14:13
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1$\begingroup$ @NateEldredge Good point. $\endgroup$– Michael GreineckerMar 18, 2018 at 14:14
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