Does the obligation to cover your hair in public apply regardless of whether men are around? For example: if you are in a friend’s apartment with only women, or if you are in an office building but having a private meeting with another woman, and there is no reason to worry that a man may walk in.
Relatedly, are these two examples considered equally ‘public’ spaces because neither is your own home, despite the fact that the second is more colloquially ‘public’ than the first?
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Questions along these lines are best addressed more personally, so that a response can take specifics of a situation and a community into account. Our response is meant as a general discussion of the relevant issues.
The obligation of head-covering in public applies no matter who is around, but who is around can affect the definition of what is public, and Chazal do not explicitly address women-only spaces in their discussions of head-covering.
Another factor in head-covering is that of dignity (as we discuss here). Dignity is less at issue where very informal dress is the norm, as at the gym. However, it is a factor in a business or professional setting, even in a meeting that is private and limited to women.
Where does this leave us? A woman typically should keep her head covered in a large gathering or in a public space, even when only women are present, because it is public and more formal. In less public or less formal all-women settings, there is more flexibility.
To take your examples, the friend’s apartment is technically a private space. Since only women are present, additional considerations of modesty do not apply. Certainly if the nature of the group does not raise additional questions of dignity, a woman can freely go uncovered there when that fits with her community’s practice.
Whether the office space is public or private depends on the situation. If others are definitely coming in and out, it is public. If other staff members might conceivably walk in, it could be considered an in-between space, like an alleyway, in which the extent of coverage can be more flexible. If the women really control access to the room or lock the door it could be considered private, though the level of formality of the meeting should be taken into account, too.
See more Q&A here.
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