Great video very intuitive! however, I still don't see why bringing in the ball of radius epsilon is really necessary - why can't you just say an open set is one that doesn't contain it's boundary and a closed set is one that does? basically just like greater than inequalities as opposed to greater than or equal to inequalities on a one dimensional line
An open interval is an open set in the real line, R. An open set in R is not necessarily an interval. It could be the union of several ('countably' many) open intervals, for example. To talk about 'openness' you need to specify the base set. An interval that is open in R is neither open nor closed in the Euclidean plane (for example.)
It's closed. If you consider it's complement you'll see that no matter how close the points get to the line, there will always exist an open ball, meaning that the complement is open, so the line is closed.
"closed is complement of open"
It is not true.
While you paint the plane in blue you didnt draw the boundary line of the plane, thats implying the plane you drew, is an open plane. It is open one side but closed on other side.
No, under any topology course I have seen, the definition of a closed set is ALWAYS the complement of an open set. The "open plane" doesn't have anything to do with it because the plane in provably both open and closed, so you could say it's actually closed "from all sides".
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