# Interior of connected metric spaces are connected?

Let $$X$$ a connected metric space. Is $$int X$$ connected?

I have a proposition that says: if $$X\subseteq Y\subseteq \overline{X}$$ and $$X$$ is connected then $$Y$$ is connected. But I'm having some problems with using this result. Because every set is open in itself we would have $$X = intX \subseteq\overline{X}$$ then $$intX$$ would be connected. Is this correct? What if $$X$$ is a connected subset of a (not necessarely connected) metric space?

• What is $int X$ when $X$ is a metric space? – xyzzyz Oct 31 '18 at 21:02
• And what is $\bar{X}$ when $X$ is a metric space? – mfl Oct 31 '18 at 21:08
• Are you sure $X$ is not a connected subset of a metric space? Otherwise, $X$ is closed so it is its own closure and $X$ is open so it is its own interior. – John Douma Oct 31 '18 at 21:10
• @JohnDouma I stated the question as it was proposed to me. I don't really know for sure. – AnalyticHarmony Oct 31 '18 at 23:03
• @xyzzyz please see my comment on Maurizio Moreschi's answer – AnalyticHarmony Oct 31 '18 at 23:13