CONTENTS
Introduction.......................................................................................................................................................... 5
I. PRELIMINARIES.............................................................................................................................................. 7
§ 1. The closures of open subsets in r. o.-equivalent topologies............................................................. 7
§ 2. The r. o.-maximal topologies.................................................................................................................... 9
§ 3. The H-closed maximal spaces................................................................................................................ 10
§ 4. R. o.-equivalence of extensions............................................................................................................... 10
§ 5. 0-continuous maps.................................................................................................................................... 11
§ 6. The Henriksen-Jerison and skeletal maps........................................................................................... 13
II. H-CLOSED EXTENSIONS OF HAUSDORFF SPACES.................................................................................... 14
§ 1. The set of 77-closed extensions of given Hausdorff space............................................................... 14
5 2. Proper maps................................................................................................................................................ 16
§ 3. Decompositions of proper maps............................................................................................................ 18
§ 4. An application to IT-closed extensions................................................................................................... 19
§ 5. The case of compact-like spaces........................................................................................................... 22
§ 6. The case of minimal Hausdorff spaces................................................................................................. 25
III. EXTREMALLY DISCONNECTED RESOLUTIONS OF HAUSDORFF SPACES................................. 26
§ 1. The set of irreducible maps onto a given Hausdorff space X............................................................ 26
§ 2. R. o.-minimal irreducible maps............................................................................................................... 30
§ 3. Extremally disconnected resolutions...................................................................................................... 31
IV. COMMUTATION OF H-CLOSED EXTENSIONS AND E. D. RESOLUTIONS...................................... 35
§ 1. Commutativity in a pullback diagram...................................................................................................... 35
§ 2. Commutativity in a pushout diagram ..................................................................................................... 37
V. PROJECTIVE AND INJECTIVE HAUSDORFF SPACES......................................................................... 39
§ 1. H-closed projective spaces. A definition and motivations.................................................................. 41
§ 2. The case of compact-like spaces........................................................................................................... 42
§ 3. Projectiveness for arbitrary H-closed spaces....................................................................................... 44
§ 4. Projectiveness for arbitrary Hausdorff spaces...................................................................................... 45
§ 5. Injective extremally disconnected spaces............................................................................................. 46
§ 6. Injective Hausdorff spaces....................................................................................................................... 48
Bibliography......................................................................................................................................................... 51