a real number as a point

Question

An element of the domain of a real-valued function of a real variable is often called a point. For example, an element (point) $p$ in the domain of a real-valued function $f$ of a real variable where $f'(p)=0$ or $f'(p)$ is undefined is called a critical point of the function. The particular type of critical point $x$ where $f'(x)=0$ is called a stationary point. As another example, "the graph $y=(x+2)(x-1)^2$ cross the x-axis at the points $x=-2$ and $x=1$." In those cases, why we called a real number as a point? Is it because we view the real numbers as points in the context of the real line?

Answer 1

The use of 'point' in this fashion goes much further than just referring to real numbers as points. The real number line (or portion of it) in the context of real analysis is an example of a space. There are many other spaces. A topological space is a far reaching generalisation of the intuitive notion of space, just to name one large class of spaces. The intuitive idea is that a space is composed of points that are somehow glued together, or are related in some way. This is a very vague concept, but it illustrates that we think of a space as a bunch of points plus some extra information (of for instance how close points are, leading to a metric space). This is why we use points in spaces. Interestingly, there is an entire area called point-free topology where one completely disposes of points. Thus there are actually spaces having no points at all, but this is an aside remark.