Examples in set-theoretic topology

We present counterexamples from various branches of set-theoretic topology.; We show in Chapter 3 that for any topological space X there is a larger space, M(X), which preserves many properties of X, and such that for any condensation (a condensation is a continuous one-to-one map onto) f : M(X) → Z, Z contains a clopen copy of X. In particular, no image of M( X) under a condensation has much better topological properties than M(X) itself.; An example of a small Tychonoff space which cannot be condensed onto any normal space is presented in Chapter 2.; Absolute countable compactness was defined by M. V. Matveev as a property lying between countable compactness and compactness. Also he defined a covering property (a) and showed that absolute countable compactness is equivalent to countable compactness plus property (a). It has been known that in most cases normality implies property (a). In particular, all the known examples of normal countably compact spaces are absolutely countably compact. In Chapter 4, we produce an example of a normal countably compact space which is not absolutely countably compact.; With every Tychonoff topological space one can naturally associate a free topological group F(X) which contains X as a subspace. It is well-known that if F( X) is normal, then all finite powers of X are also normal. In Chapter 5, we present an example showing that the converse of this fact is not true under the assumption of the Continuum Hypothesis.; All presented results are new and answer published questions of A. V. Arhangel'skii, I. Juhasz and Z. Szentmiklossy, O. G. Okunev and O. V. Sipacheva, M. G. Tkachenko, S. Watson.