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We present an example of an algebra that is generated by ${\omega }_1$ elements, and cannot be made a topological algebra. This answers a problem posed by W. Żelazko.

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_k\left(X\right)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$-space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_k\left(X\right)$ is Ascoli iff $C_k\left(X\right)$ is a $k_{ℝ}$-space iff X is locally compact. Moreover, $C_k\left(X\right)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and...