A note on tameness of families having bounded variation

We show that for arbitrary linearly ordered set $X$ any bounded family of (not necessarily, continuous) real valued functions on $X$ with bounded total variation does not contain independent sequences. We obtain generalized Helly’s sequential compactness type theorems. One of the theorems asserts that for every compact metric space $(Y,d)$ the compact space $B{V}_r(X,Y)$ of all functions $X\to Y$ with variation $\le r$ is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space $M_{+}(X,Y)$ of all order preserving maps $X\to Y$ is sequentially compact where $Y$ is a compact metrizable partially ordered space in the sense of Nachbin.