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Currently displaying 1 – 2 of 2

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An example of a non-topologizable algebra

R. Frankiewicz; G. Plebanek — 1995

Studia Mathematica

We present an example of an algebra that is generated by $ω_{1}$ elements, and cannot be made a topological algebra. This answers a problem posed by W. Żelazko.

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan; J. Kąkol; G. Plebanek — 2016

Studia Mathematica

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_{k} (X)$ 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 $C_{k} (X)$ is Ascoli iff $C_{k} (X)$ is a $k_{ℝ}$ -space iff X is locally compact. Moreover, $C_{k} (X)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and...

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