© 1996, Topology Atlas

Let C_p(X) be the space of all continuous real-valued functions on a Tychonoff space X with the pointwise convergent topology. The symbol ~ stands for linear homeomorphism. For two spaces X and Y are said to be l-equivalent if C_p(X) ~_l C_p(Y) holds. X ~_l Y means that X and Y are l-equivalent. In 1980, Pavlovskií [PV] showed that, for two finite polyphedra X and Y, X ~_l Y holds if and only if dim X coincides with dim Y. Subsequently, the classification problem of certain classes of spaces up to l-equivalence has been studied by several general topologists. For zero-dimensional separable metrizable spaces, some exact results in this direction have been known (see [BG]). Below, the results for non-zero dimensional spaces in this direction, which I know are listed. The symbols R and Dⁿ denote the real line and the n-disk respectively.

1. A. N. Dranishnikov, 1986 [DR]
   If U is an open subset of Rⁿ then U ~_l the topological sum of countable infinitely many Dⁿ's ~_l Rⁿ.
2. A. Koyama and T. Okada, 1987 [KO]
   If X is a 1-dimensional compact ANR with finite ramification points or a continuum which is a one-to-one continuous image of [0, infinity) then X ~_l D¹.
3. Arhangel'skí, 1989 [AR]
   If X is a compact CW complex with dim X = n > 0 then X ~_l Dⁿ
4. Valov, 1991 [VA]
   If K is a compact subset of Rⁿ with dim K = n then K ~_l Dⁿ
5. Valov, 1991 [VA]
   Let E = Rⁿ, I^omega (= the Hilbert cube), µⁿ (=n-dim. universal Menger compactum) or R^omega and suppose X is a separable metrizable space which is an E-manifold.
   • If X is compact and E= I^omega or µⁿ then X ~_l E
   • If X is not compact then X ~_l the topological sum of countable infinitely many E's.
6. K. Kawamura and K. Morishita, [KM]
   If X is a compact topological manifolds with dim X = n then X ~_l Dⁿ
7. K. Kawamura and K. Morishita, [KM]
   Let X be a (non-compact) CW complex.
   • If dim X = n and 1 ¾ n then C_p(X) ~ ½_iC_p(Dⁱ)^tau_i(X)
   • If dim X = infinity then C_p(X) ~ ½_iC_p(Dⁱ)^tau_i(X) x ½_iC_p(Dⁱ)^alpha_i(X) where;
     • tau_i(X) is the cardinality of the set of all i-cells of X which are not contained in the closure of any other cell of X (if tau_i(X) is finite, we may suppose tau_i(X) = 1 by (3)) and
     • alpha_i(X) is the cardinality of the set of all i-cells of X which are contained in the closures of infinitely many cells of X.

References

[AR] A. V. Arhangel'skí, On linear topological classification of spaces of continuous functions in the topology of pointwise convergence, Math. Sbornik 70 (1991), pp. 129-142.

[BG] J. A. Baars and J. A. M. de Groot, On topological and linear equivalence of certain function spaces, CWI Tract 86 (1992).

[DR] A. N. Dranishnikov, Absolute F-valued retracts and spaces of functions in the topology of pointwise convergence, Siberian Math. J. 27 (1986), pp. 366-376.

[KM] K. Kawamura and K. Morishita, Linear topological classification of certain function spaces on manifolds and CW complexes, to appear in Top. Appl..

[KO] A. Koyama and T. Okada, On compacta which are l-equivalent to Iⁿ, Tsukuba J. Math. 11 (1987), pp. 147-156.

[PV] D. S. Pavlovskí, On spaces of continuous functions, Soviet Math. Dokl. 22 (1980), pp. 34-37.

[VA] V. M. Valov, Linear topological classifications of certain function spaces, Trans. Amer. Math. Soc. 327 (1991), pp. 583-600.