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In this paper it is proved that a k-dimensional closed subset Xc Rn admits a projection p into one of the coordinate k-planes such that dimp(X)=k. The purpose of this note is to prove the following theorem: THEOREM. Let Xc RN be a k-dimensional compact subset of Rn, l Ril x ... x Ri. The question of whether the above statement is true was raised by J. D. Lawson in connection with a problem concerning n-dimensional topological semilattices on a Peano continuum. I am indebted to J. Nagata for bringing it to my attention. It is well known that a subset Yc Rk is k-dimensional if and only if it has nonempty interior, Int Y# 0 (see, e.g., [1, Theorem IV.3, p. 44]). Consequently, the theorem can be given the following equivalent form: THEOREM. Let Xc Rn be a k-dimensional compact subset of Rn, 1 2, and we prove it for n. Let Xc Rn, dim X=k, k<n. Consider any of Received by the editors December 13, 1972 and, in revised form, February 16, 1973. AMS (MOS) subject classifications (1970). Primary 54F45; Secondary 54E45, 54F05, 22A30.

Compact subsets of 𝑅ⁿ and dimension of their projections