One-to-one continuous mappings of topological spaces

Compact condensations of Hausdorff spaces

“…Raukhvarger [31], V.V. Proizvolov [25], A.S. Parhomenko [22,23], Y.M. Smirnov [34], N. Hadzhiivanov [36], V.K.…”

“…The fact that X \ Y can be condensed onto a compactum for every countable Y was established for metrizable compacta by Raukhvarger [31], for products of metrizable compacta by Proizvolov [25], for diadic compacta by Belugin [10], for weakly diadic compacta (including polyadic and centered spaces) by Kulpa and Turzanski [17], for zero-dimensional first countable compacta by Belugin [11]. On the other hand Ponomarev [6] proved that if we remove from the remainder ω * = βω \ ω of the Čech-Stone compactification of ω, a countable subset D then ω * \ D has no condensation onto a compactum.…”

Quasicompactness and functionally Hausdorff spaces

“…They observed that the pz s q axiom lies strictly between the Hausdorff and completely regular axioms, and that it neither implies nor is implied by the T 3 axiom. In [7] Proizvolov touched upon topological spaces with this property and called them functionally Hausdorff spaces.…”

Mappings onto the plane

“…In [11], V. V. Proizvolov claimed to have proved that if Xis connected, locally compact, and paracompact then/must be a homeomorphism. Later [12], he used this result to show that if X is connected, locally connected, and locally compact then/is a homeomorphism.…”

“…Later [12], he used this result to show that if X is connected, locally connected, and locally compact then/is a homeomorphism. There was, however, an error in the proof given in [11], and examples have been given by Kenneth Whyburn [20] and L. C. Glaser [7], [8], and [9] which show that neither of the above theorems is valid when «^3.…”