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Keywords:
function space; pointwise convergence; linearly ordered topological space; Lindelöf space; Cantor tree
function space; pointwise convergence; linearly ordered topological space; Lindelöf space; Cantor tree
Summary:
A.V. Arkhangel'skii asked that, is it true that every space $Y$ of countable tightness is homeomorphic to a subspace (to a closed subspace) of $C_p(X)$ where $X$ is Lin\-de\-löf? $C_p(X)$ denotes the space of all continuous real-valued functions on a space $X$ with the topology of pointwise convergence. In this note we show that the two arrows space is a counterexample for the problem by showing that every separable compact linearly ordered topological space is second countable if it is homeomorphic to a subspace of $C_p(X)$ where $X$ is Lindelöf. Other counterexamples for the problem are also given by making use of the Cantor tree. In addition, we remark that every separable supercompact space is first countable if it is homeomorphic to a subspace of $C_p(X)$ where $X$ is Lindelöf.
References:
[1] Arkhangel'skii A.V.: Problems in $C_p$-theory. in: J. van Mill and G.M. Reed, Eds., {Open Problems in Topology}, North-Holland, 1990, 601-615. MR 1078667 | Zbl 0994.54020
[2] Engelking R.: General Topology. Sigma Series in Pure Math. 6, Helderman Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[3] Lutzer D.J.: On generalized ordered spaces. Dissertationes Math. 89 (1971). MR 0324668 | Zbl 0228.54026
[4] Mill J. van: Supercompactness and Wallman spaces. Mathematical Centre Tracts 85 (1977). MR 0464160
[5] Mill J. van, Mills C.F.: On the character of supercompact spaces. Top. Proceed. 3 (1978), 227-236. MR 0540493
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