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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"? – JavaMan Jun 8 '11 at 15:55
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. – Asaf Karagila Jun 8 '11 at 15:59
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:…… – Martin Sleziak Jun 8 '11 at 16:01
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 $L^1$. – Robert Bell Jun 8 '11 at 17:01
To complement what Robert said: Claim 2 becomes true if integrals are taken over finite intervals: $\int_a^b\lvert f(x)\rvert^p\, dx$. – Giuseppe Negro Jun 9 '11 at 1:52

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