Great and succinct. Found this vid after about an hour of struggling with various other resources online and understand the concept in 8 minutes. Cheerios!
I think it's possible to compress 00:00 to 03:08 -- "Set M of metric space R is open - if it consisting entirely of interior points" . I think in the previous video you have already covered all details. But in any case - thank you. The vision of such detail can be lie in that this video by itself is self-contained.
awesome video mate but i have a question, what’s stopping us from setting the radius of the epsilon ball to a number big enough so that part of the circle was outside the set meaning that the point was included in an open set?
open balls are usually written with dotted lines right? I see that you have not used a dotted line for the open ball in the video. Please guide me with this. @Ben1994
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.
The set is open if every point in the set has a neighbourhood within the set itself (for example the interval (0, 1), every x within has a neighbourhood, just pick the lesser between _x_ and _1-x_ (think of it geometrically, a line from 0 to 1 but not including those ones) and set e lesser than min{x, 1-x}. You see that (x-e, x+e) is within (0,1) for an e, foe every x (you can stretch the proof by formalising). A closed set is a set which its complementar is open.
"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".
[a, b] is closed since R - [a, b] is open (-inf, b) U (a, inf)= R - [a, b] (-inf, b) is open (a, inf) is open the union of two open sets outcomes another open set
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.)
