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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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    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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