Is it permissible to walk on Shabbat in a wheelchair or with a cane in a public area that is not encircled by an Eruv?

According to Halacha, a person who is, Heaven forbid, incapable of walking without a wheelchair may wheel himself in a public domain on Shabbat.  In such a case, the wheelchair is considered part of the person's body, insofar as he cannot move about without it.  However, although he is permitted to wheel himself in a public area, other people may not push his wheelchair.  And when the individual wheels himself through a public domain on Shabbat, he must ensure not to carry any other objects.  Thus, for example, he may not keep his Tallit, Siddur or eyeglasses on his lap or next to him on the seat as he wheels himself.

This principle applies to canes, as well.  If a person is incapable of walking without a cane, then Halacha considers the cane his "third leg," and he may therefore use it for walking in a public domain on Shabbat.  If, however, a person uses the cane merely for additional comfort and support, but he is capable of walking without it, then walking with a cane is tantamount to carrying the cane and is therefore forbidden in a public domain on Shabbat.  Canes differ in this regard from eyeglasses, which one may wear in a public domain on Shabbat even if he can see without them.  Since glasses are worn, and not carried, they become like a person's clothing.  A cane, by contrast, is carried, rather than worn, and therefore one may use it in a public domain on Shabbat only if he cannot walk without it, in which case it serves as an additional leg.  (Or L’sion, Helek 2, Perek 23:5, and see Menuhat Ahava, Helek 3, page 354.)

Of course, these rules apply only in public areas without an Eruv.  In public areas encircled by a valid Eruv, all carrying is permissible, and one is certainly allowed to walk with a wheelchair or cane under all circumstances.

Summary: If a person cannot walk without a wheelchair or cane, he may use a wheelchair or cane while walking in a public area on Shabbat, even if there is no Eruv.  In the case of a wheelchair, he must ensure that no objects are with him in the wheelchair, and only he is permitted to wheel the chair; others may not wheel him through the public domain.