# The basis criterion (topology)/Statement

From Maths

Grade: A

This page requires references, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.

The message provided is:

The message provided is:

I know introduction to topological manifolds has 'em

## Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]\mathcal{B}\in\mathcal{P}(\mathcal{P}(X))[/ilmath] be a topological basis for [ilmath](X,\mathcal{ J })[/ilmath]. Then^{[1]}:

- [ilmath]\forall U\in\mathcal{P}(X)\big[U\in\mathcal{J}\iff\underbrace{\forall p\in U\exists B\in\mathcal{B}[p\in B\subseteq U]}_{\text{basis criterion} }\big][/ilmath]
^{[Note 1]}

If a subset [ilmath]U[/ilmath] of [ilmath]X[/ilmath] satisfies^{[Note 2]} [ilmath]\forall p\in U\exists B\in\mathcal{B}[p\in B\subseteq U][/ilmath] we say it satisfies the *basis criterion* with respect to [ilmath]\mathcal{B} [/ilmath]^{[1]}

## Notes

- ↑ Note that when we write [ilmath]p\in B\subseteq U[/ilmath] we actually mean [ilmath]p\in B\wedge B\subseteq U[/ilmath]. This is a very slight abuse of notation and the meaning of what is written should be obvious to all without this note
- ↑ This means "if a [ilmath]U\in\mathcal{P}(X)[/ilmath] satisfies...

## References

- ↑
^{1.0}^{1.1}Introduction to Topological Manifolds - John M. Lee