Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined by
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00017831/resource/name/S0008414X00017831_eqn01.gif?pub-status=live)
A subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such that
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00017831/resource/name/S0008414X00017831_eqn02.gif?pub-status=live)
and the dimension of σ is defined to be (n — m). Let
denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.