11:30 , could you please explain why we can't use the sets (-infinity, 0) and (0, infinity) to prove the dis-connectivity of rationals ? why is their union not covering the rational set ?
Thanks for paying close attention! This was a small mistake- the definition at the very start of the video should require that A and B are PROPER subsets. So this will explicitly disallow letting one be empty. Sorry for my oversight!
Very nice. One comment I'd like to put forward (if you care for it) is that the original topological definition, in my view, is much more revealing and easier to swallow even if one's only talking about (subsets of) ℝ or ℝⁿ; viz. a topological space X is disconnected if it is homeomorphuc to the disjoint union (coproduct) Y ∐ Z of two non-empty topological spaces Y and Z (empty doesn't count for given any topological space X, one has X ≅ X ∐ ∅, and as such will define a trivial concept); equivalently, X = U ∪ V for two disjoint non-empty open (or closed, therefore) subsets U and V (images of Y and Z resp.); equivalently, X has a non-empty proper subset U which is both open and closed, ...etc.for X ⊆ ℝⁿ (or any other parenting space), the definition applied to the subspace topology is easily seen to specialize to the given definition. 😊 Good work again. I too have been trying to create video courses to lead up to algebraic geometry (hopefully all the way to the GAGA theorems).. for quite some time now, in fact. And, I know how hard it is.. my best wishes, please do keep it up. 👍
Yes I do feel a bit dissatisfied with defining separations in the way I did. But the alternative as you suggest involves discussing the subspace topology, which is cumbersome in a course which otherwise has no need mention of it.
@@ChrisStaecker just glanced over it. well, um an alternative I forgot to include before would be the definition: A (⊆ ℝⁿ) is separated if there is a surjective continuous function f : A -> {0,1}. We can't avoid continous functions, can we? All the caution that is needed is to note that these needn't extend to all of ℝ (or ℝⁿ). A continuous function f : A -> B (⊆ ℝ, say) could be defined as a function such that whenever a sequence {x_n} in A converges to x (in A), f(x_n) converges to f(x). That's innocent enough, isn't it? [So, we can check that if A ⊆ ℝ misses a point c between a, b ∈ A, it is disconnected. Once the converse is established, and since the definition immediately gives that a continuous image of a connected set is connected, the intermediate value theorem follows, which is the point here anyway.]
11:30 , could you please explain why we can't use the sets (-infinity, 0) and (0, infinity) to prove the dis-connectivity of rationals ? why is their union not covering the rational set ?
Their union would exclude 0, which is rational.
What prevents one of A or B being the original set and the other being the empty set?
Thanks for paying close attention! This was a small mistake- the definition at the very start of the video should require that A and B are PROPER subsets. So this will explicitly disallow letting one be empty. Sorry for my oversight!
Very nice. One comment I'd like to put forward (if you care for it) is that the original topological definition, in my view, is much more revealing and easier to swallow even if one's only talking about (subsets of) ℝ or ℝⁿ; viz. a topological space X is disconnected if it is homeomorphuc to the disjoint union (coproduct) Y ∐ Z of two non-empty topological spaces Y and Z (empty doesn't count for given any topological space X, one has X ≅ X ∐ ∅, and as such will define a trivial concept); equivalently, X = U ∪ V for two disjoint non-empty open (or closed, therefore) subsets U and V (images of Y and Z resp.); equivalently, X has a non-empty proper subset U which is both open and closed, ...etc.for X ⊆ ℝⁿ (or any other parenting space), the definition applied to the subspace topology is easily seen to specialize to the given definition.
😊 Good work again. I too have been trying to create video courses to lead up to algebraic geometry (hopefully all the way to the GAGA theorems).. for quite some time now, in fact. And, I know how hard it is.. my best wishes, please do keep it up. 👍
Yes I do feel a bit dissatisfied with defining separations in the way I did. But the alternative as you suggest involves discussing the subspace topology, which is cumbersome in a course which otherwise has no need mention of it.
@@ChrisStaecker I see... hmm.. but, I thought you were following Rudin, weren't you?
@@nnaammuuss I'm using "Understanding Analysis" by Abbott.
@@ChrisStaecker just glanced over it. well, um an alternative I forgot to include before would be the definition: A (⊆ ℝⁿ) is separated if there is a surjective continuous function f : A -> {0,1}. We can't avoid continous functions, can we? All the caution that is needed is to note that these needn't extend to all of ℝ (or ℝⁿ).
A continuous function f : A -> B (⊆ ℝ, say) could be defined as a function such that whenever a sequence {x_n} in A converges to x (in A), f(x_n) converges to f(x). That's innocent enough, isn't it?
[So, we can check that if A ⊆ ℝ misses a point c between a, b ∈ A, it is disconnected. Once the converse is established, and since the definition immediately gives that a continuous image of a connected set is connected, the intermediate value theorem follows, which is the point here anyway.]