Space Suggestion
This is $X = \{x\in\mathbb{R}^2 : |x|\leq 1\}$ where $|\cdot|$ is any norm on $\mathbb{R}^2$. Topologically those are all the same.
Rationale
In general there is a closed $n$-ball, but I think adding $2$-ball is significant because $1$-ball to $2$-ball is quite different topologically. This should be reflected by a lot of properties here on pi-base (of course all of $n$-balls are important topologically, but this might not be apparent from the standard properties considered on pi-base).
Relationship to other spaces and properties
There's a lot of properties this classical space satisfies, too much to write in here.
Space Suggestion
This is$X = \{x\in\mathbb{R}^2 : |x|\leq 1\}$ where $|\cdot|$ is any norm on $\mathbb{R}^2$ . Topologically those are all the same.
Rationale
In general there is a closed$n$ -ball, but I think adding $2$ -ball is significant because $1$ -ball to $2$ -ball is quite different topologically. This should be reflected by a lot of properties here on pi-base (of course all of $n$ -balls are important topologically, but this might not be apparent from the standard properties considered on pi-base).
Relationship to other spaces and properties
There's a lot of properties this classical space satisfies, too much to write in here.