The Approximate Variation of Univariate Uniform Space Valued Functions and Pointwise Selection Principles

Given a Hausdorff uniform space $$X$$ with the countable gage of pseudometrics uniformity , we introduce concept approximate variation function $$f$$ mapping subset $$T$$ reals into : this is infimum family Jordan-type variations all functions $$g:T\to X$$ which differ from in each pseudometric, generated by pseudometric gage, not greater than $$\varepsilon>0$$ . We prove following compactness theorem topology pointwise convergence: if relatively sequentially compact sequence such that limit superior its finite for and then it contains subsequence converges on domain to bounded regulated (in generalized sense). illustrate result appropriate sharp examples present new characterization valued terms variation.