I enjoyed the video but I just wanted to point out that you can generalize the definition of dense sets to non-metric topological spaces. A subset M in a topological space (X, τ) is dense in X iff for every x in X and open subset U in τ containing x, there exists an m in M such that m is also in U. The concept of distance isn't required and avoids the need for the ball with radius epsilon. Another more succinct way to say it, is that the closure of M is X. So every point in X is either in M or on the boundary of M. If M has this property, then M is dense in X.
Thank you very much for this video! It helped me a lot, but there are a few things I would like to point out: I don't understand this definition of a dense set from your first illustration to be honest. In that illustration, it looks like it would be very easy to find a a radius r for which the ball around x does not contain any element of M. Also this is a little nitpicking, but you are saying "there is always a rational number between any two real numbers you pick". Rational numbers are real numbers themselves, so it would maybe be clearer if you said something along the line of "No matter which real number you pick, it's either a rational number itself, or there is a rational number right next to it". I think that is a correct statemement, somebody correct me please if I'm wrong.
Hello, I remember that you had a video on Nowhere Dense sets, but I cannot seem to find it anymore. I am doing a research paper on this topic, and I would very much appreciate it if you could give me the link to this video. Keep up the great work!
If you take M=X={0}, then M is dense in X since the closure of M is equal to X. However, there exists an element of X, namely 0, which is NOT a limit point of M since no epsilon balls around 0 contain a point of M distinct from 0, which contradicts your definition of M being dense in X. Could you please explain this?
A dense set does not need to be countable. The rationals in his example just happened to be. For example: every uncountable set X is dense within itself. That makes X an uncountable dense subset of X.
I enjoy every minute of your videos! You are great! God bless you! Keep making advanced math videos.
I enjoyed the video but I just wanted to point out that you can generalize the definition of dense sets to non-metric topological spaces. A subset M in a topological space (X, τ) is dense in X iff for every x in X and open subset U in τ containing x, there exists an m in M such that m is also in U. The concept of distance isn't required and avoids the need for the ball with radius epsilon.
Another more succinct way to say it, is that the closure of M is X. So every point in X is either in M or on the boundary of M. If M has this property, then M is dense in X.
many thanks ! awesome explanation - really helps visualizing it.
simple yet intuitive explanation! Thank you sir.
Excellent explanation!
How will you always get a open ball around x for all r which has a Point of M??
Thanks for your video, it was verry helpful for me
Thank you very much for this video! It helped me a lot, but there are a few things I would like to point out:
I don't understand this definition of a dense set from your first illustration to be honest. In that illustration, it looks like it would be very easy to find a a radius r for which the ball around x does not contain any element of M.
Also this is a little nitpicking, but you are saying "there is always a rational number between any two real numbers you pick". Rational numbers are real numbers themselves, so it would maybe be clearer if you said something along the line of "No matter which real number you pick, it's either a rational number itself, or there is a rational number right next to it". I think that is a correct statemement, somebody correct me please if I'm wrong.
Not all heroes wear capes
great explanation sir.
Hello, I remember that you had a video on Nowhere Dense sets, but I cannot seem to find it anymore. I am doing a research paper on this topic, and I would very much appreciate it if you could give me the link to this video. Keep up the great work!
Great use of notation
dude, you are amazing!!!!!!! do you teach in college?
Great example
If you take M=X={0}, then M is dense in X since the closure of M is equal to X. However, there exists an element of X, namely 0, which is NOT a limit point of M since no epsilon balls around 0 contain a point of M distinct from 0, which contradicts your definition of M being dense in X. Could you please explain this?
M is dense if M union D(M) is X.
I.e., if M closure is equal to X.
M is dense in X if closure(M) is a proper superset of X
No its dense if the closure IS X.
Thank you!
how can we prove dence set is countable?
matric space may be uncountable
A dense set does not need to be countable. The rationals in his example just happened to be. For example: every uncountable set X is dense within itself. That makes X an uncountable dense subset of X.
thanks! That's helpful
love your accent
great video but fix your microphone pls
My book gives definition X=M closure