## The Convergence Approach to Exponentiable Maps (2000)

Venue: | 352 MARIA MANUEL CLEMENTINO, DIRK HOFMANN AND WALTER |

Citations: | 9 - 7 self |

### BibTeX

@INPROCEEDINGS{Clementino00theconvergence,

author = {Maria Manuel Clementino and Dirk Hofmann and Walter Tholen},

title = {The Convergence Approach to Exponentiable Maps},

booktitle = {352 MARIA MANUEL CLEMENTINO, DIRK HOFMANN AND WALTER},

year = {2000},

pages = {139--160}

}

### OpenURL

### Abstract

Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilter-interpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the classical result for compact Hausdorff spaces. Furthermore, in generalization of the Whitehead-Michael characterization of locally compact Hausdorff spaces, we characterize exponentiable maps of Top between Hausdorff spaces as restrictions of perfect maps to open subspaces.

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Citation Context ...e U ) x instead of X (U) ! x. It turns out that Pisani's characterization allows for a natural generalization from spaces to maps, which occured to us after seeing the Janelidze-Sobral criterion (see =-=[19]-=- and [5]) for triquotient maps of finite topological spaces in the sense of Michael [25]. Hence, we first looked at the category PrSet of preordered sets (= reflexive transitive graphs = sets with a r... |

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Citation Context ...s of these theorems. In particular, we give refined versions and generalizations of the invariance and inverse invariance theorems of local compactness under perfect mappings, as first established by =-=[21]-=- and [35] and recorded in [10]. Finally, coming back to our discussion of exponentiable spaces, we study in Section 5 the map-version of the Whitehead-Michael characterization of exponentiable spaces ... |

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Citation Context ...d in the full subcategory of sober locally compact spaces. Proof. (1) Factor f (in any finitely-complete category) as X X \Theta Y Y - ! 1 X ; f ? - p Y where both factors are exponentiable (see [6], =-=[33]-=-). (2) Apply the categorical version of (1) to Top=Z in lieu of Top. (3), (4) follow from [33], Corollary 3.4(3) and Proposition 3.6, respectively. We also mention that Theorem B makes it easy to prov... |

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Citation Context ...nancial assistance by NSERC. 1 contains a smaller one with the property that every open cover of the given neighbourhood contains a finite subcover of the smaller one. Generalizing Whitehead's result =-=[36]-=- for Hausdorff spaces, Brown [3] already in 1964 showed that locally compact spaces (in which every point has a base of compact neighbourhoods) are exponentiable. For Hausdorff spaces the two notions ... |

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Citation Context ...nstead of X (U) ! x. It turns out that Pisani's characterization allows for a natural generalization from spaces to maps, which occured to us after seeing the Janelidze-Sobral criterion (see [19] and =-=[5]-=-) for triquotient maps of finite topological spaces in the sense of Michael [25]. Hence, we first looked at the category PrSet of preordered sets (= reflexive transitive graphs = sets with a reflexive... |

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Citation Context ...ness obviously amounts to Hausdorffness of X in this case. Categorically it is clear that Theorem A is the "right" map generalization of the space result of the 1940s (see [6]). Recent work =-=by Pisani [30]-=- (as well as our own experience in trying to characterize effective descent and triquotient maps in Top, see [31] and [7]) suggested to us that the language of convergence would facilitate a proof of ... |

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Citation Context ...sbell [22]). For an elementary account of these results, see [16]. Trading now Top for the category Top/Y of spaces X over the fixed base space Y , given by continuous maps f : X → Y , Niefield [31]=-=, [32]-=- gave an elegant but, when put in standard topological terminology, generally complicated topological characterization of exponentiable maps in Top, which entails the Day-Kelly result in case Y = 1 is... |

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Citation Context ...ven for sober spaces (Hofmann-Lawson [16]). There is no known constructive example of an exponentiable space that is not locally compact (Isbell [17]). For an elementary account of these results, see =-=[11]-=-. Trading now Top for the category Top=Y of spaces X over the fixed base space Y , given by continuous maps f : X ! Y , Niefield [26], [27] gave an elegant but, when put in standard topological termin... |

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Citation Context ...rization of exponentiable spaces X in Top is as follows. Let UX be the set of ultrafilters on X, and for U 2 UUX, let X (U) = [ A2U " a2A a 2 be the sum of the ultrafilters a (a 2 A ` UX, A 2 U);=-= see [14]-=-. Now X is exponentiable if and only if X has the ultrafilter interpolation property: whenever X (U) ! x in X, then there is a 2 UX with U ! a and a ! x (with a naturally defined notion of convergence... |

2 |
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Citation Context ...arated and locally closed and therefore exponentiable (see [26]), and so are perfect maps, by Theorem A. Furthermore, exponentiable and separated maps are closed under composition. 5.6 Remark. James (=-=[18]-=-, p.58) gives the construction of the fibrewise Alexandroff compactification, which provides for every continuous map f : X ! Y a factorization f = (X X + Y ) - js\Theta - q with an open embedding j a... |

2 |
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2 |
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1 |
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Citation Context ...an just use the pullback (5) in Set and apply 2.9. Hence, every morphism in URS is exponentiable, i.e., URS is locally cartesian closed. 2.11 (Coincidence of partial products in Top and in PsTop, see =-=[4]-=-). An inspection of the proof 2.10 yields the known fact that also the subcategory PsTop is locally cartesian closed (see [15]), and that the partial products are formed as in URS. 8 For f : X ! Y and... |

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1 |
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Citation Context ...sbell [17]). For an elementary account of these results, see [11]. Trading now Top for the category Top=Y of spaces X over the fixed base space Y , given by continuous maps f : X ! Y , Niefield [26], =-=[27]-=- gave an elegant but, when put in standard topological terminology, generally complicated topological characterization of exponentiable maps in Top, which entails the Day-Kelly result in case Y = 1 is... |

1 |
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Citation Context .... Under suitable restrictions on X and Y it becomes very applicable as well; for instance, it shows that every map from a locally compact space to a locally Hausdorff space is exponentiable (Niefield =-=[28]-=-). However, we have not been able to derive from it the statement we were aiming for, namely: Theorem A Every perfect map of topological spaces is exponentiable in Top. Here we call a continuous map f... |

1 |
Exponentiable maps and triquotients in Top (preprint
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(Show Context)
Citation Context ... subspace embedding, in which case exponentiability of f means local closedness of X in Y (so that X is open in its closure X in Y ), and even when f is just an injective map, as was shown by Richter =-=[32]-=-. Under suitable restrictions on X and Y it becomes very applicable as well; for instance, it shows that every map from a locally compact space to a locally Hausdorff space is exponentiable (Niefield ... |

1 |
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(Show Context)
Citation Context ...e theorems. In particular, we give refined versions and generalizations of the invariance and inverse invariance theorems of local compactness under perfect mappings, as first established by [21] and =-=[35]-=- and recorded in [10]. Finally, coming back to our discussion of exponentiable spaces, we study in Section 5 the map-version of the Whitehead-Michael characterization of exponentiable spaces as locall... |

1 |
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Citation Context ...wn (and trivial), see [31]. � Remark. Proper surjective maps are biquotient maps, i.e., pullback-stable quotient maps (see [29]). As was noted by the anonymous referee (as well as in the recent pape=-=r [9]), state-=-ments (1) and (3) of 4.2 can be generalized considerably by trading “proper and surjective” for “biquotient”. The proof of this generalization is in fact purely categorical if one uses the wel... |

1 |
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Citation Context ...topological spaces, i.e., those X ∈ URS satisfying (1) and (1 1 � 2 ) a → x whenever b ⊆ a (see [35]). b→x For a categorical analysis of the first two in the chain of bireflective embeddings=-= see also [20]. Top → PrTo-=-p → PsTop → URS, 2.5 (Prime Filter Theorem, see [25]). Recall that a filter of a 0-1-lattice is an up-closed subset F ⊆ L which is a sub-semilattice of (L, ∧, 1); it is prime if 0 �∈ F , a... |

1 |
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Citation Context ...e theorems. In particular, we give refined versions and generalizations of the invariance and inverse invariance theorems of local compactness under perfect mappings, as first established by [26] and =-=[40]-=- and recorded in [15]. Finally, coming back to our discussion of exponentiable spaces, we study in Section 5 the map-version of the Whitehead-Michael characterization of exponentiable spaces as locall... |