It is well known that the metric topology  on interval space is not compatible with the interval inclusion monotonicity property in the sense that there may exist monotonic functions which are not continuous and conversely. This paper provides a quasi-metric topology for the interval space consistent with the real line topology and whose continuous functions are exactly the monotonic ones. The provided quasi-metric is not a metric only because it fails to satisfy the symmetrical property. The quoted title is due to the fact that except to the Hausdorff property of the metric-which does not fit for our point of view-the other good metric properties remain.
Reliable Computing – Springer Journals
Published: Oct 14, 2004
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