Point-set topology forms the foundation for much of topology. In engineering mathematics, we’re only really concerned with definitions.
Regions and sets
A neighbourhood of is any disc for some . We can think of it as more or less some small region around the point. In , a neighbourhood of is a ball.
Assume we have some region .
- A point is an interior point if it is possible to find a disc centred at those points that will lie entirely in . Intuitively, this point cannot be on the border of the region.
- Similarly, a point is a boundary point if every neighbourhood of has at least one point of and one point not in . The set of all boundary points of is called the boundary of .
- Remark: a boundary point of need not be in . For example, if we define the unit square in without its extremities, then any boundary point will be on its extremities but not within the set.
- For a set that includes a value approaching infinity, infinity isn’t part of the boundary since you can’t get points outside of the set at infinity.
Regions (or sets) are open if they consist entirely of interior points (or equivalently if it contains none of its boundary points). Closed regions are the union of all boundary points and open regions. Sets can be neither open nor closed.
An open set is connected if , there is a path joining and consisting of finitely many segments lying entirely in . More intuitively, a set is non-connected if there’s a discontinuity (i.e., a connected set should be in one piece). An open connected set is a domain.
A curve is simple if it doesn’t intersect itself (except possible at end points). A domain is simply connected if any simple closed curve in can be continuously shrunk to a point in while staying in .
Intuitively, D is simply connected if it has no holes.