Contents Introduction.................................................................................................................... 5 § 1. BlSC-subbases..................................................................................................... 6 § 2. The families D(U) and C(U) and the spaces $X_U$ for a given U.............. 7 § 3. A lemma on chains in maximal BISC-subbases of continua...................... 9 § 4. Further properties of the spaces $X_U$.......................................................... 12 § 5. The Main Lemma.................................................................................................. 14 § 6. Application to the characterization of several cubes...................................... 16 $ 7. Topological characterization of n-spheres....................................................... 17 References.................................................................................................................... 29