Dimension under Dense Linear Mappings of Function Spaces

All topological spaces are assumed to be Tichonoff. For a space $X$, by $dim(X)$ we denote the covering dimension of the space $X$. For a topological space $X$, let $C_p(X)$ denote the space of real continuous functions on the space X endowed with the pointwise convergence topology.

Let $\kappa$ be an infinite cardinal number. A normal space $X$ is called strongly $\kappa$-dimensional if it is a union of $\kappa$ many closed finite dimensional spaces. For $\kappa=\omega$, one obtains the well known class of strongly countable dimensional spaces. A space is called $\kappa$-compact if it is a union of $\kappa$ many its compact subspaces.

We improve results concerning invariance of dimension-like properties under transformations of function spaces from [Za]. In particular we show that for a normal, strongly $\kappa$-dimensional space $X$ and a normal, metacompact, locally $\sigma$-compact space $Y$ if there exists a continuous linear operator $T:C_p(X)\xrightarrow[]{}C_p(Y)$ with dense image, then $Y$ is strongly $\kappa$-dimensional as well. The finite-dimensional case is examined as well. We also obtain a compactification theorem that may be of independent interest: every normal strongly $\kappa$-dimensional space admits a strongly $\kappa$-dimensional $\kappa$-compactification. Recall that for a space $X$, we have $dim\,(X)=dim\,(\beta X)$ and that there exist a normal, strongly countable dimensional space without strongly countable dimensional compactification [EP,Example 5.5].

1. [EP] R. Engelking, E. Pol, Countable-dimensional spaces: a survey, Dissertationes Math. 216, (1983).

2. [Za] K. Zakrzewski, Function spaces on Corson-like compacta, Results Math. 80, 75 (2025).