In London, where not everyone holds of the Eruv (although it as acknowledged by many to be halachically acceptable), can a woman going to the Mikveh on Friday night wear a shower cap underneath her Sheitel so that on the way back home from the Mikveh, when her hair is dripping wet, her sheitel does not get ruined, and it also has the benefit of protecting her from catching a cold.
It is permitted to wear the shower cap under the sheitel.
The Shulchan Aruch (301:11) discusses a case of wearing a key as a tachshit, an item of jewelry. Of course, one wishes to take the key for the purpose of opening the front door, but in order to permit transporting it on Shabbos, one uses it as an item of jewelry. The ruling of the Shulchan Aruch is that it is permitted in principle, but forbidden because of “ro’in,” meaning because of others who will see the key, and think that it is permitted to carry on Shabbos. The Rema adds that some permit the practice, and that this is the common custom.
The swimming cap appears to be of similar nature to the key. One wears it on one’s body, and its basic intention is to protect the body (to protect hair from becoming wet), although the intention is to use it for protecting the sheitel. For the shower cap, there is no concern for people seeing it, and therefore it appears to be permitted.
Although there is room to argue that the principle purpose of the key is decorative (as the Mishnah Berurah writes), the same can be said of the shower cap (the principle purpose is for wearing it on one’s hair), and it would therefore be permitted for wear.
A similar decision emerges from the matter of taking towels to the bathhouse. The Gemara (Shabbos 147b) rules that it is permitted to carry towels to the bathhouse by wearing them on one’s head. The purpose is only to carry them, but because one wears them, no prohibition is involved.
The Meiri explains that this is not the same as the question of “porephes” (Shabbos 65a), in which the Gemara questions whether it is permitted for a woman to carry a nut into the public domain (for her son to play with) by using it as a button, and remains undecided (the halachah is therefore that it is permitted to do so in a karmelis, but not in a full public domain–as ruled by the Shulchan Aruch 303:23). The case of the towel is more lenient, because the towel is actually worn on the body in a normal way of wearing an item of clothing.
These Gemaros do not appear to concur with the ruling of the Yerushalmi, as cited by the Mishnah Berurah in his Biur Halachah (concerning watches). There is room, however, to draw a distinction between the above cases, and the case of the Yerushalmi, which speaks of a composite item in which the ring is worn as an ornament, but the stamp for a purpose. I will not elaborate on this matter here.
Therefore, the case of the shower cap, which is actually worn on the head in a manner of wearing an item of clothing (one “wears” a swimming cap, and one doesn’t “carry it” on one’s head), is similar to the case of the towel (and the case of the key), and it is thus permitted to wear the shower cap, even if one’s intention is to protect the sheitel.
The fact that the shower cap is not usually worn in public does not negate its status as an item of clothing, for its purpose in protecting hair is sufficient for this. Rav Moshe Feinstein (Orach Chaim 1:111, concerning a wrist watch) writes that there is an advantage (in terms of carrying on Shabbos) for items that is worn on the body itself (such as a watch), and in our case the item is not only worn on the body, but is actually an item of clothing, and therefore permitted.
The fact that there is possibly an eiruv, and that the shower cap is (also) being used to prevent cold (this might depend on the season), join in affirming the heiter.