What is "x" in this definition of a basis?

[Hill]

TL;DR

The definition in the book appears to be missing something.

In this definition,

what is $x$ in the property (ii)?

Did they mean, "(ii) For any open $V \subset X$ and any $x \in V$, there exists some $U \in \mathscr{B}$ with $x \in U$ and $U \subset V$?
Or something else?

 

[mathwonk]

yes they meant what you said. however this is a strange definition, as property 3 follows from property 2, so is not needed. The book seems to be confusing two different characterizations. Namely given a topology on X, one wants to define a base for that topology, which is properties 1 and 2.
Then given just a set X, without a topology, one also wants to define when a collection of subsets of X is a base for some topology, which is property 3 plus the property, unstated there, that X is the union of the subsets in the base. see the Wikipedia article on base of a topology, for a correct discussion. given this evidence, I would suggest possibly getting a different book.

 

[Hill]

yes they meant what you said. however this is a strange definition, as property 3 follows from property 2, so is not needed. The book seems to be confusing two different characterizations. Namely given a topology on X, one wants to define a base for that topology, which is properties 1 and 2.
Then given just a set X, without a topology, one also wants to define when a collection of subsets of X is a base for some topology, which is property 3 plus the property, unstated there, that X is the union of the subsets in the base. see the Wikipedia article on base of a topology, for a correct discussion. given this evidence, I would suggest possibly getting a different book.

Thank you.

I see what you mean. But this is unexpected because the book is this:

The Math You Need: A Comprehensive Survey of Undergraduate Mathematics,
Thomas Mack,
©2023 Massachusetts Institute of Technology

 

[Hill]

yes they meant what you said. however this is a strange definition, as property 3 follows from property 2, so is not needed. The book seems to be confusing two different characterizations. Namely given a topology on X, one wants to define a base for that topology, which is properties 1 and 2.
Then given just a set X, without a topology, one also wants to define when a collection of subsets of X is a base for some topology, which is property 3 plus the property, unstated there, that X is the union of the subsets in the base. see the Wikipedia article on base of a topology, for a correct discussion. given this evidence, I would suggest possibly getting a different book.

Following your suggestion, I've got a different book:
"All the Math You Missed But Need to Know for Graduate School", Thomas A. Garrity, 2021.

At least, they have the definition right:

 

[mathwonk]

I think your new book is probably a good choice.