My question is going to focus on quite a counterintuitive thing.
A couple of preliminaries. I understand geometric space as a set of points. A point, in turn, is an abstract idealization of an exact position in a space. There is a point in each position of geometric space.
While exploring the possible ways to define a line, I have come up with the definition of adjacent points. I classify two non-coincident points of geometric space adjacent if there are no points between them. Intuition hints there is no problem with such definition.
However, it turns out that it's more complex than I have thought. A point is a position in a space, thus it's possible to assign a unique number to each point. It's well-known that smallest strictly positive rational number does not exist (see here for a proof) which implies that there will always be a point between two distinct points.
Now here comes the question: do adjacent points somehow exist? If not, then how is it possible that geometric space is infinitely dense in points but has no adjacent points as there is always a point between any two points?
I'd like to shed some light on the topic. I believe it's known enough.
The study of the properties of spaces based on points being near each other is known as topology.
When this concept of points being near each other can be expressed as a real number (i.e. distance), we say the space is a metric space and that that metric specifies a notion of closeness(a.k.a a topology) on that space. Most spaces that match geometric intuition are metric spaces. A topology which is constructed by a metric is known as a metrizable topology.
The basic intuition that you have is that different points shouldn’t be infinitely close to each other. There are many ways of turning this intuition into a rigorous mathematical statement. These various ways are known as the separation axioms (https://en.m.wikipedia.org/wiki/Separation_axiom). Depending on your exact formulation, some of these ways are stronger than others, so you can put them in order and give names to topologies that satisfies these. One of the most common of these are if a space is Hausdorff ($T_2$). A Hausdorff space is one where any two different points have neighborhoods of each that are disjoint from each other.
The Niagara Smirnov metrization theorem (https://en.m.wikipedia.org/wiki/Nagata–Smirnov_metrization_theorem) gives an important classification of metrizable topologies - highlighting that one of the main parts of intuition of what you might call a ‘geometric space’ that it is regular Hausdorff (i.e. a notion of points being non adjacent to each other) and has a countable locally finite basis (sorta like it not being infinite dimensional nor too big in size).