My undergraduate real analysis class is talking about set connectedness and has already discussed the fact that a set $E\subset X\subset Y$ is compact in $X$ iff it is compact in $Y$. Is the same true for connectedness? Is connectedness an intrinsic property of a set?
Is connectedness an intrinsic property of a set?
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real-analysis
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1Just for fun, I'll stress that we really need $E \subset X$ here. For example, $(0,1)$ is not compact in $\mathbb{R}$, but it is compact in (the relative topology on) $\{ 1/2 \}$. While a lazy person might try to say "compactness is hereditary", it isn't quite true. (Not that you have made such a claim. I'm just musing out loud.) – 2012-10-24
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You mean that compactness/connectedness is an intrinsic property of the topology.
And yes, this is true. The way to show this depends on what definition of connected you're using. For example, any continuous function to a discrete space being constant characterizes connected spaces. Using this definition, we don't refer to ambient spaces at all, so the property must be intrinsic.