Clone Segments of the Tychonoff Modification of Space

Representations of algebraic theories by continuous maps

“…These three sections thus form the 'topological' part of the paper. Moreover, the spaces F(a { ) and F(a 2 ) constructed in Sections 4 and 5 are metrizable, and this provides a joint strengthening of those results of [8,9] and [6] which concern clone segment equality and isomorphism.…”

“…The fact that CR satisfies (a) and (b) is well-known. To see that CR satisfies also (c), we let JXf be determined by all Hausdorff spaces X in which the closure of every neighborhood of any point contains its cozero neighborhood (see [8], where such spaces are said to have the R-property). Define a space X o on the set Q n [0,1] of all rationals in the closed interval [0,1] whose Euclidean topology is changed at 0 e X o so that the system n To, i ) \ T | n = 1, 2,... J , where T = U | k = 1, 2,... J is its local open basis there.…”

“…If X is a C-semirigid space with card(P \ C) > 3 and X k its fc-th power, then/ (P k ) c C for any continuous map / : X k -*• X other than a constant or a projection, see [9]. (b) In [8], a metric Q on P was called extremally C-semirigid (for a set C C P) if…”

“…Hausdorff spaces X in which the closure of every neighborhood of any point contains its cozero neighborhood (see [8], where such spaces are said to have the R-property). …”

The Clone of a Topological Space: an inspiring book by Walter Taylor

“…The idea of [38] was to 'tighten' the construction by means of finding representing spaces such that Y is obtained from X through a modification of a prescribed kind. In this, [38] succeeded for the following modifications: the completely regular (that is, Tychonoff) modification, compactly generated modification, sequential modification, and some others. In [30], these results were enriched by the simultaneous representation of the equality ∼ 1 and isomorphism ∼ 2 of clone segments (in the sense of 4.2).…”