Connected and Path Connected Sets, Real Analysis II

In this video, we explore two topological properties for sets in a metric space: connectedness and path-connectedness. These are distinct concepts, with connectedness being defined by what it is not (a set is connected if it cannot be separated by two disjoint open sets), and path-connectedness being more intuitive (a set is path-connected if any two points in the set can be joined by a continuous path that lies within the set). (We show a continuous function preserves these properties here: • Connected and Path Con... )
First, we define connected sets and demonstrate this by showing that the set of rational numbers is disconnected. This is done by finding two open sets that cover the rational numbers without overlap. Then, we prove that the closed interval [a,b] on the real number line is connected by setting up a contradiction argument to show that no such pair of open sets can separate the interval.
Next, we define path-connected sets, which are easier to visualize. A set is path-connected if any two points in it can be connected by a continuous path that lies within the set. We demonstrate examples of path-connected sets, like a straight line segment or a semicircle in the Euclidean plane. Importantly, we prove that if a set is path-connected, it must also be connected, although the converse is not necessarily true.
Finally, we explore an example of a set that is connected but not path-connected: the topologist’s sine curve. This famous example shows how a set can be connected yet still lack the ability to trace continuous paths between certain points. We use two different explanations to show that the sine curve cannot be path-connected but is nonetheless connected, as it cannot be separated by open sets.
#Topology #RealAnalysis #AdvancedCalculus #MetricSpaces #Mathematics #math #maths