Meaning and concept of space

[Jignesh77]

Good morning,

I hope you're well and thank you in advance. I got Masters in Chemistry (1998) and studied maths till 12 grade(1994). I love learning maths in free time.
What are the different meanings of the term "space" in mathematics in simple words?

 

[BigBeachBanana]

That depends on the context. Can you be more specific?

 

[Jignesh77]

Thank you for your reply.
The first meaning of "space" that we learn in geometry class in primary school.
I also see that the term used as in vector space and metric space which I didn't learn in high school.
I just need some intuitive explanation and simple explanation to understand the concept.
English isn't my first language so words like "space" and "structure" in this context( may be technical in this sense) hamper my understanding of certain concepts related to these topic.

 

[Dr.Peterson]

Thank you for your reply.
The first meaning of "space" that we learn in geometry class in primary school.
I also see that the term used as in vector space and metric space which I didn't learn in high school.
I just need some intuitive explanation and simple explanation to understand the concept.
English isn't my first language so words like "space" and "structure" in this context( may be technical in this sense) hamper my understanding of certain concepts related to these topic.

Does this help at all?

Space (mathematics) - Wikipedia

en.wikipedia.org

If not, what is lacking?

 

[Jignesh77]

Thank you!
"In mathematics, a space is a set (sometimes known as a universe) with a definition (structure) of relationships among the elements of the set."


I don't understand "with a definition (structure) of relationships among the elements of the set."
Is the definition we learn in elementary algebra, physical space, the notion of 2d and 3d , different?

 

[blamocur]

Thank you!
"In mathematics, a space is a set (sometimes known as a universe) with a definition (structure) of relationships among the elements of the set."


I don't understand "with a definition (structure) of relationships among the elements of the set."
Is the definition we learn in elementary algebra, physical space, the notion of 2d and 3d , different?

It depends on the space. Look at the definitions, for example, of topological space and vector space.
I do not believe the word "space" itself means much, but only when used with an adjective like "topological" or "vector". Otherwise it is just a set, but don't ask me about the meaning of "set",

 

[Dr.Peterson]

Thank you!
"In mathematics, a space is a set (sometimes known as a universe) with a definition (structure) of relationships among the elements of the set."


I don't understand "with a definition (structure) of relationships among the elements of the set."
Is the definition we learn in elementary algebra, physical space, the notion of 2d and 3d , different?

The term is intentionally vague, I think, allowing for anything broadly analogous to the physical (3D Euclidean) space we are familiar with. This space is a set of points, related (structured) by coordinates or distances, and it's the model for what we mean by "space".

So if we are thinking about some sort of set that is more than just a set of unrelated points, we can call it a "____ space", with the adjective labeling either the kind of relationships involved. or the kind of objects in the space.

In order to understand the first sentence, which you quoted, you have to move on to the next sentence. (This is often true of understanding anything new! Read on for examples and explanations.) Here, the next sentence is:

While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.​

This gives examples that you can look into (some of them easier to understand than others), and also that important fact that the notion is vague -- it doesn't really have a definition at all. So don't expect to ever have a clear idea of exactly what a space is; it can be just about anything.

Taking some of the links, such as the one for "structure" may also help. Again, I'd be mostly looking for more examples. Also, as usual with Wikipedia, don't think you have to read it all; the first few paragraphs, and the starts of some examples, are probably all you need to see before being overwhelmed.

 

[Jignesh77]

Thank you!

 

[fresh_42]

Space is usually a short-term term for what in real life would be an environment. It describes the set of things where the specific part of mathematics takes place.

Spaces are intentionally minimal in the sense that certain spaces often have additional properties beyond the one that defines them. A vector space is a set of points that is closed under addition and subtraction, compressions and stretches. Other structures like for instance algebras begin with being a vector space but we can additionally multiply vectors which we call the elements from a vector space. A metric space is a set where we can measure distances. A topological space is roughly speaking a set where points can have neighborhoods such that we can have the concept of continuous mappings.

Spaces don't exclude each other. A Euclidean space like the flat plane is a vector space (a set of arrows that can be added, subtracted, stretched, and compressed), but it is also a geometric space where most of the theorems of Euclid's Elements take place, a metric space because we have distances, an inner product space which we use to measure angles, a topological space because we can draw and define continuous functions.

You see from the discussion here that "space" alone isn't a well-defined mathematical term. We need an additional adjective or noun to specify which kind of space we are talking about.

Here is another description of possible "spaces"

Learn the Basics of Hilbert Spaces and Their Relatives | Physics Forums

Hilbert spaces can look rather different, and which one is used in certain cases is by no means self-evident. To refer to Hilbert spaces...

www.physicsforums.com

explained differently than the Wikipedia article linked above.