Is the empty space really zero dimensional?

[felixpernegger]

Zero dimensional is defined as having a basis of cloped sets; trivially the empty space has this property. However, the name comes from the inductive dimension (which ought to be mentioned in the description of the property anyhow).

But the empty space has inductive dimension -1 , not zero.

Personally, I would keep the empty space zero dimensional but write about this inconsistency in the description of Zero dimensional

[Moniker1998]

Sure, we did something like this for https://topology.pi-base.org/properties/P000217

[prabau]

Multiple sources not specializing in dimension theory define zero dimensional as we did. We can keep it that way and mention it corresponds to $\dim X\le 0$ for suitable notions of dimension.

[Moniker1998]

@prabau small inductive dimension, denoted $\text{ind}$ by most sources. The notation $\text{dim}$ is usually used for covering dimension or something similar

[prabau]

Yes. What I was trying to get at is that the same is true for multiple notions of dimension. The zero-dimensional property of pi-base is equivalent to small inductive dimension ind X <= 0, and also equivalent to covering dimension dim X <= 0.

At least, I think that's right. Can you confirm?

[Moniker1998]

@prabau off the top of my head, that's incorrect. The latter would be equivalent to ultranormal or strongly zero-dimensional depending on definition