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ehrenfest
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Homework Statement
Urysohn's lemma
My book says that the "if" part of Urysohn's lemma is obvious with no explanation. Can someone explain why?
HallsofIvy said:It would have been a good idea to actually state Urysohn's lemma as it is given in your book. Sometimes statements vary from one book to another. In particular you should note that Urysohn's lemma only applies in NORMAL spaces. What is the definition of a "Normal" topological space?
[0, 1/2) is not an open subset of Rehrenfest said:Sorry. I meant to put a link to Wikipedia, which has the same statement of Urysohn's Lemma as that in my book.
http://en.wikipedia.org/wiki/Urysohns_lemma
It comes down to whether the sets [0,1/2) and (1/2,1] are open. Apparently this is obvious to other people, but it seems counterintuitive to me because I thought open sets were open intervals.
Urysohn's Lemma is a theorem in topology that states that for any two disjoint closed sets in a normal topological space, there exists a continuous function that separates them.
The "if" part of Urysohn's Lemma states that if a topological space is normal, then it satisfies the conditions of the lemma.
Urysohn's Lemma is significant because it helps prove other important theorems in topology, such as the Tietze Extension Theorem and the Urysohn Metrization Theorem.
One example of how Urysohn's Lemma is used is in the proof of the Urysohn Metrization Theorem, which states that any second-countable, Hausdorff, regular space is metrizable. Urysohn's Lemma is used to construct a metric on the space by defining a function that separates points in the space and then using this function to define a metric.
Yes, Urysohn's Lemma only applies to normal topological spaces and cannot be used to prove the separation of disjoint closed sets in non-normal spaces.