What is a rigorous definition for "subspace"?

Question

I learned that the subspace of $\mathbb R^2$ can only be the origin or a $\mathbb R$ space that pass through the origin.

I also learned that the subspace of $X$ is any subset that contains an open set.

The two definitions seem to be contradictory.

My question is, what is rigorous definition for subspace?

Answer 1

My question is, what is rigorous definition for subspace?

The word "subspace" on its own doesn't mean much and depends on the context. It can mean:

1. Topological subspace: any subset with subspace topology
2. Metric subspace: any subset with metric restricted to it
3. Linear subspace: a subset closed under addition and scalar multiplication
4. ...

and more. One could use the same term for normed, Banach, inner product, Hilbert, affine, measurable, probability subspaces. Even schemes or topoi. The list is just huge, close to endless.

All those have precise definitions, often unrelated to other structures. And so you need to understand the context, in order to understand what "subspace" means.

I learned that the subspace of $\mathbb R^2$ can only be the origin or a $\mathbb R$ space that pass through the origin.

Here, the author most likely refers to linear subspace. Note that $\mathbb{R}^2$ itself is also a linear subspace of $\mathbb{R}^2$.

I also learned that the subspace of $X$ is any subset that contains an open set.

Unlikely anyone defined "subspace" like that. In the context of metric and/or topology, the term "subspace" literally means any subset. Subset that contain an open subset are typically called a subset with nonempty interior.

Answer 2

Being a subspace depends on some kind of space of which you can be a subspace. Therefore there are very many different definitions of subspaces depending on your setup.

You can take sub (vector) spaces of $\mathbb{R}^2$ which gives you (up to you not mentioning the entire space as a subspace of itself) the subspaces that you mentioned at first.

If you consider $\mathbb{R}^2$ as a topological/metric space, you can also consider (topological/metric) subspaces of it. These will once again be topological/metric spaces and do not have to be sub vector spaces at all.

There are nevertheless very general definitions of what it means to a "subsomething", but I would not suggest you to think about this level of generality at this point.