Connectedness of A={$(x,y):x \in \mathbb{I}\,$ or$ \, y \in \mathbb{I}$}

Asked Jun 03 '20 at 20:43

Active Jun 03 '20 at 20:43

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i have the following problem about connectedness, prove that in the Euclidean plane

A={$(x,y):x \in \mathbb{I}\,$ or$ \, y \in \mathbb{I}$} is connected

(This is my interpretation of the exercise, since the statement says: “show that the set of points on the plane with at least one irrational coordinate is a connected set in the usual metric.”)

I have tried in several ways, but it causes me a problem, that some of the coordinates have to be irrational

asked Jun 03 '20 at 20:43

Haus

It really depends on what $\mathbb{I}$ is. – copper.hat Jun 03 '20 at 20:45
@copper.hat - the OP makes it clear that it stands for the irrational numbers – uniquesolution Jun 03 '20 at 20:46
The set of numbers irrationals – Haus Jun 03 '20 at 20:46
You could also use $\mathbb{Q}^c$. – copper.hat Jun 03 '20 at 20:47
The complement of a countably infinite subset of $\mathbb{R}^{2}$ is always path connected. Therefore, $A$ is path connected. – Johnny El Curvas Jun 03 '20 at 20:49

Connectedness of A={$(x,y):x \in \mathbb{I}\,$ or$ \, y \in \mathbb{I}$}

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