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Hello everyone I am looking for a couple of references:

Claim 1 states that an open and connected set in $R^n$ is path-connected. Or more general an open, connected and locally connected set is path-connected.

Claim 2 states that $L^p_{BC}$ is a subset of $L^1_{BC}$, where $L^p_{BC}$ is the set of continuous and bounded functions such that $\int_{\mathbb{R}} |f(x)| dx < \infty$

Thanks in advance for any help.

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    Have you tried Munkres's "Topology"?2011-06-08
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    Open and connected imply locally connected (trivially, if I might add), I believe you meant for open, connected and locally path connected in Claim 1.2011-06-08
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    As far as the firtst claim is concerned, searching google books for: *open connected "path connected" euclidean* leads to two books by Stephen Krantz containing the claim: http://books.google.com/books?id=O3tyezxgv28C&pg=PA27&dq=open+connected+%22path+connected%22+euclidean&hl=en&ei=g5zvTYq4HYehOrPyiN4B&sa=X&oi=book_result&ct=result&resnum=2&ved=0CC4Q6AEwAQ#v=onepage&q=open%20connected%20%22path%20connected%22%20euclidean&f=false http://books.google.com/books?id=LUhabKjfQZYC&pg=PA33&dq=open+connected+%22path+connected%22+euclidean&hl=en&ei=g5zvTYq4HYehOrPyiN4B&sa=X&oi=book_result2011-06-08
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    @Martin S.- Thanks a lot. It was very easy to find with this link, and I will definitely use one of them.2011-06-08
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    Claim 2 is false. Let $f(x) = |1/x|$ for $|x| \geq 1$ and $f(x) = 1$ for $|x| \leq 1$. Then $f$ is continuous and bounded, belongs to $L^p$ for all $p > 1$, but does not belong to $L^1$.2011-06-08
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    To complement what Robert said: Claim 2 becomes true if integrals are taken over finite intervals: $\int_a^b\lvert f(x)\rvert^p\, dx$.2011-06-09

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