@@Sorya-gf7qw If x is rational, the union of A and B are not the whole set of rational numbers Q, because the rational number x is missing. So for the argument in 7:15, you need that x is irrational.
great video! Do you have a video on locally connectedness? There seem to be these interesting sets that are connected but not locally path connected? I am not familiar with the english terminology, but basicly there is this set, where a component can't be seperated as an open subset, but you can't have a path leading to it at the same time. It's kind of weird but also super fascinating.
A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. But seems that [1,2]U[3,4] is disconnected but also cannot be the union of two nonempty open sets.
Real analysis is the study of spaces of real numbers \R^n, including properties of subsets of \R^n and real-valued functions on them. The most elementary form of real analysis is real-valued calculus, which students often study in high school. At the college level, it includes vector calculus and express courses on real analysis, and it dovetails into fields like measure theory and functional analysis.
Additionally, it is also the study of the properties of different sets such as compactness, closure, completeness and many more. This then extends into a topic known as metric spaces which in turn generalises to topological spaces.
Sure it does. if a,b are both in {5}, then both a=5 and b=5, so the interval (a,b) is the empty set, and the supposition that (if c in (a,b), then c in E) is vacuously true, as (a,b) is empty.
