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**Covariant and contravariant points of view in topology with applications to function spaces.**
*(English)*
Zbl 0936.54022

The author introduces what he calls covariant and contravariant topologies and studies these primarily in the context of function spaces. A covariant topology on \(X\) is determined by a family of maps from spaces to \(X\) – it is the largest topology that makes these maps continuous; quotient and sum topologies arise in this manner. Similarly, a contravariant topology is the smallest that makes all members of a family of maps from \(X\) to spaces continuous, examples are the subspace and product topology.

The covariant point of view gives rise to a known (but possibly not so well known) topology on the set \(C(X,Y)\) of continuous maps from \(X\) to \(Y\). One takes the class of all maps \(f:S\to C(X,Y)\) for which the map \(\widehat f:S\times X\to Y\), defined by \(\widehat f(s,x)=f(s)(x)\), is continuous. One obtains the largest proper topology on \(C(X,Y)\) [R. Arens and J. Dugundji, Pac. J. Math. 1, 5-31 (1951; Zbl 0044.11801)]; the author calls it the basic covariant topology. By combining both viewpoints one can obtain the compact-open topology and the topology of pointwise convergence: Take the class of all maps \(f:K\to X\) where \(K\) is compact (or finite) and endow \(C(X,Y)\) with the contravariant topology induced by the maps \(f^*:C(X,Y)\to C(K,Y)\) and the basic covariant topology on the spaces \(C(K,Y)\); one obtains the compact-open topology (the topology of pointwise convergence) in this way. In the second part of the paper the author considers topologies on \(X\times Y\) that ensure that the natural map from \(C(X\times Y,Z)\) to \(C(X,C(Y,Z))\) will be a homeomorphism in case both function spaces carry the compact-open topology or the topology of pointwise convergence.

Reviewer’s remarks: The covariant and contravariant topologies are not new inventions; they appear for example in the textbooks of B. von Querenburg [Mengentheoretische Topologie (1979; Zbl 0431.54001)] or of W. Gähler [Grundstrukturen der Analysis (1977; Zbl 0346.54001)]. The author’s Proposition 1.8 is erroneous: it is the sequential spaces rather than the Fréchet spaces, that carry the covariant topology with respect to a family of maps whose domain is the converging sequence.

The covariant point of view gives rise to a known (but possibly not so well known) topology on the set \(C(X,Y)\) of continuous maps from \(X\) to \(Y\). One takes the class of all maps \(f:S\to C(X,Y)\) for which the map \(\widehat f:S\times X\to Y\), defined by \(\widehat f(s,x)=f(s)(x)\), is continuous. One obtains the largest proper topology on \(C(X,Y)\) [R. Arens and J. Dugundji, Pac. J. Math. 1, 5-31 (1951; Zbl 0044.11801)]; the author calls it the basic covariant topology. By combining both viewpoints one can obtain the compact-open topology and the topology of pointwise convergence: Take the class of all maps \(f:K\to X\) where \(K\) is compact (or finite) and endow \(C(X,Y)\) with the contravariant topology induced by the maps \(f^*:C(X,Y)\to C(K,Y)\) and the basic covariant topology on the spaces \(C(K,Y)\); one obtains the compact-open topology (the topology of pointwise convergence) in this way. In the second part of the paper the author considers topologies on \(X\times Y\) that ensure that the natural map from \(C(X\times Y,Z)\) to \(C(X,C(Y,Z))\) will be a homeomorphism in case both function spaces carry the compact-open topology or the topology of pointwise convergence.

Reviewer’s remarks: The covariant and contravariant topologies are not new inventions; they appear for example in the textbooks of B. von Querenburg [Mengentheoretische Topologie (1979; Zbl 0431.54001)] or of W. Gähler [Grundstrukturen der Analysis (1977; Zbl 0346.54001)]. The author’s Proposition 1.8 is erroneous: it is the sequential spaces rather than the Fréchet spaces, that carry the covariant topology with respect to a family of maps whose domain is the converging sequence.

Reviewer: K.P.Hart (Delft)

### MSC:

54B99 | Basic constructions in general topology |

54C35 | Function spaces in general topology |

54B05 | Subspaces in general topology |

54B10 | Product spaces in general topology |

54B15 | Quotient spaces, decompositions in general topology |

54D50 | \(k\)-spaces |

54D45 | Local compactness, \(\sigma\)-compactness |