Geodesic
Projective Elements in Categories with Perfect θ-Continuous Maps
Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 872-884

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In 1958 Gleason [6] proved the following :THEOREM. In the category of compact Hausdorff spaces and continuous maps, the projective elements are precisely the extremally disconnected spaces.The projective elements in many topological categories with perfect continuous functions as morphisms have been found since that time. For example: In the following categories the projective elements are precisely the extremally disconnected spaces: (i) The category of Tychonov spaces and perfect continuous functions. [4] [11]. (ii) The category of regular spaces and perfect continuous functions. [4] [12]. (iii) The category of Hausdorff spaces and perfect continuous functions. [10] [1]. (iv) In the category of Hausdorff spaces and continuous k-maps the projective members are precisely the extremally disconnected k-spaces. [14]. In 1963 Iliadis [7] constructed for every Hausdorff space X the so called Iliadis absolute E[X], which is a maximal pre-image of X under irreducible θ-continuous maps. 
Vermeer, Hans; Wattel, Evert. Projective Elements in Categories with Perfect θ-Continuous Maps. Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 872-884. doi: 10.4153/CJM-1981-068-6
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