To begin with, when we are talking about “space”, Even though, the general sense is about “area or emptiness.” It can fall into various senses. If you are in the field of art such as an architecture or an artist, your space is meant to be the physical area or emptiness surrounding us or on the piece of work where you can fill your ideas in. If you are in Psychology, it can mean the personal space which is the comfort territory surrounding you that makes you feel safe. In the non-physical sense, it is the personal attitude towards the minimum interaction that others can do to oneself. In Science, we try to give definitions and properties on abstract things, especially in Physics and Maths, to make them more concrete in order that we can understand their nature and potentially make use out of them.
In Physics which uses the definition from Maths to describe things, space comes usually with time, the so-called “space and time” or “spacetime.” There is a clear definition from the general relativity theory that a spacetime is “a four dimensional Lorentizian manifold .” So now, what is “manifold” ? If you search further, you will find more terms like “topology” which will make you more confused. The manifold is a topological space, informally, a manifold is a space that modeled on Euclidean space .
One of the easiest actions that you can do when you face new things in your life and you are curious to understand them is that you should try to link them with the old or foundation knowledge of yourself. In my case when I met this word for the first time and without proper educational lecture, I read through most of the information available online starting from the spaces that I was familiar with which were “Euclidean space” and “Vector space”. Then, I tried to link and expanded them further to more terms like “Tangent space” and to higher abstract levels such as “Hilbert space”, “Banach space”, “Inner product space” and their superset which was “Normed vector space or Lebesgue space.” Sooner, I got to know “Metric space” and its superset which was our “Topological space.” Finally, above all, it was “set” which was the well-known term again.
With limited knowledge at that time (and for now) to give the mathematical definition to all related aforementioned spaces, I could only summarize those spaces in one figure which at least made me understand more and be able to classify those things easier than before. Whenever I encounter those keywords again, this figure will suddenly appear in my mind to support me in dealing with what I have confronted. The simple key take-away message to describe and remember those spaces in mathematical sense is that it is “the set of something with certain rules/conditions/properties on it”.
Topology from Maths is a big deal in various fields e.g. Physics and Computer Science. It plays an important role as a basic concept ranging from many applications in imaging processing and computer vision to sophisticated natural phenomena as gravitational wave. If you have time, it is interesting and quite a challenging task trying to understand it more. Consequently, you will understand the nature deeper.