Does my adult child live with me if they live in an apartment leased to me separately from my house and I paid all their living expenses?

No, they do not live with you if they are in their own apartment.  But if they are a full-time student, living away at school is a temporary absence and you can still say they live with you.    Or if not a student, whether your adult child lives with you may not matter for the purpose of claiming them as a qualified relative dependent.   Do they meet the criteria of having less than $5050 of income not including Social Security?    Did you pay for over half their support?

WHO CAN I CLAIM AS A DEPENDENT?

You can claim a child, relative, friend, or fiancé (etc.) as a dependent on your 2024 tax return as long as they meet the following requirements:

Qualifying child

• They're related to you.
• They aren't claimed as a dependent by someone else.
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They're under the age of 19 (or 24 for full-time students).
• No age limit for permanently and totally disabled children.
• They lived with you for more than half the year (exceptions apply).
• They didn't provide more than half of their own support for the year.

Qualifying relative

• They don't have to be related to you (despite the name).
• They aren't claimed as a dependent by someone else.
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They lived with you the entire year (exceptions apply).
• They made less than $5050 in 2024.   (Not counting Social Security)
• You provided more than half of their financial support.

When you add someone as a dependent, we'll ask a series of questions to make sure you can claim them. There may be other tax benefits you can get when you claim a dependent.

Related Information:

• What does "financially support another person" mean?

No, but a dependent child may not need to live with you to be your "Qualifying Relative".

In this case their support and income would be a factor. 

According to the IRS:

For Qualifying Relative:

"Relatives who don't have to live with you. A person related to you in any of the following ways doesn't have to live with you all year as a member of your household to meet this test.
Your child, stepchild, or foster child, or a descendant of any of them (for example, your grandchild). (A legally adopted child is considered your child.)
Your brother, sister, half brother, half sister, stepbrother, or stepsister.
Your father, mother, grandparent, or other direct ancestor, but not foster parent.
Your stepfather or stepmother.
A son or daughter of your brother or sister.
A son or daughter of your half brother or half sister.
A brother or sister of your father or mother.
Your son-in-law, daughter-in-law, father-in-law, mother-in-law, brother-in-law, or sister-in-law.
Any of these relationships that were established by marriage aren't ended by death or divorce."
 

Qualifying relative

A qualifying relative must meet general rules for dependents and pass these tests:

• Not a qualifying child: Isn't your qualifying child or the qualifying child of any other taxpayer
• Member of household or relationship: Lives with you all year as a member of your household or is a specific type of relative
• Gross income: Has gross income under $5,050
• Support: Gets more than half their financial support from you

See the full rules for a qualifying relative”

@robert209