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The answer is no for your second question if we assume sufficiently strong large cardinal hypotheses. Let $X$ be a set of measurable cardinality and then give $X$ the discrete uniformity. Then $X$ is …

I am unsure if the following result is exactly what you wanted, but I did it anyways since it seemed like a fun exercise to do. $\mathbf{Theorem}$ Let $(X,\mathcal{U}),(Y,\mathcal{V}),(Z,\mathcal{W}) …

The definition of uniform normality given is not a property of the underlying topology of your uniform space, but the definition of uniform normality is a property of the underlying proximity on your …

Here is a result that completely characterizes the uniformities that are generated by metrics as opposed to pseudometrics. $\mathbf{Theorem}$ Let $(X,\mathcal{U})$ be a uniform space. Then the follow …

I shall first give an example of an open set in a proximity space that is not measurable. Let $X$ be an uncountable discrete space with the proximity $\delta$ induced by the one-point compactification …

It is very common for a topological space to be a complete uniform space in some uniformity, but it is less common for a topological space to satisfy the Baire category theorem since the proof of the …