Let $K=[0,1]\times \{0,1\}$ be endowed with the topology arising from the lexicographic order on it. It is known that $K$ is compact, Hausdorff, first-countable and perfectly normal. Furthermore, the space $c_0[0,1]$ is a quotient of $C(K)$. Is $c_0[0,1]$ a (complemented) subspace of $C(K)$?
$c_0[0,1]$ in $C(K)$
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general-topology
functional-analysis
banach-spaces
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0In Fabian-Habala et all the space $K$ is called "two arrow space", see [p.634](http://books.google.com/books?id=5BDX2NNsqR4C&pg=PA634). You can find there also the result about the quotient, which is mentioned in the question. – 2011-11-28
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0Thanks. They prove that the dual $C(K)^*$ is weak*-separable, but the dual of $c_0[0,1]^*$ is not, hence $c_0[0,1]$ does not embed into $C(K)$. Is that right? – 2011-11-28