Introduction ..................................................... 1
1 Preliminaries ................................................. 5
1.1 Functions, clones, and preordered sets ................... 5
1.2 Topological spaces ...................................... 10
1.3 Topological relational structures ....................... 19
2 A General Galois Theory for Continuous Operations and
Closed Relations ............................................. 21
2.1 Some basic concepts ..................................... 21
2.2 Clones of continuous operations ......................... 24
2.3 Clones of closed relations .............................. 27
2.4 The Galois connection cPol-cInv ......................... 30
3 A Relational Localisation Theory for Topological Algebras .... 33
3.1 Finding suitable subsets ................................ 34
3.2 Neighbourhoods .......................................... 40
3.3 The local topological algebra A|υ ....................... 45
3.4 Local isomorphisms ...................................... 50
3.5 The topology N(A) ....................................... 56
3.6 Covers .................................................. 61
3.7 Full covers ............................................. 68
3.8 Irreducibility .......................................... 71
4 Topological Quasivarieties and the Localisation Functor ...... 75
4.1 Topological quasivarieties .............................. 75
4.2 Natural polymorphisms ................................... 82
4.3 Natural neighbourhoods and covers ....................... 89
4.4 The localisation functor ................................ 95
5 Compactness Reasoning for Topological Algebras .............. 107
5.1 Idempotents in compact Hausdorff semigroups ............ 108
5.2 Operational compactness ................................ 114
5.3 Minimality ............................................. 130
5.4 Contractivity .......................................... 137
6 Studying Some Examples ...................................... 143
6.1 Essentially unary structures .......................... 143
6.2 Topological lattices ................................... 147
6.3 Topological modules .................................... 162
Acknowledgements ............................................... 179
Index of Notation .............................................. 181
Bibliography ................................................... 189
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