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Topology of metric spaces S. Kumaresan
Publisher: Alpha Science International, Ltd

I have some topology notes here that claim that on any metric space (A,d), A is an open set. Topology as a structure enables one to model continuity and convergence locally. This section was created so that the movement from metric spaces to topological spaces can be seen as a larger jump than the one from Euclidean spaces to metric spaces. And what does it mean for spaces which are sufficiently nice, like metric spaces?" Let's state the result just so we're all on the same page. But surely we can just take a closed set and define a metric on it, like [0,1] in R with normal metric? So is Cauchiness a metric property? I am assuming that the reader is familiar with the terms metric, metric space, topological space, and compact set. Be a compact metrizable space and Y a metrizable space. Is it that a property is metric if it is related to the metric used on the space. That's how in the same space like R, we can prove that cauchiness is not topological by changing the metric. Here's my more modern topological interpretation of this claim. Math in Plain English: Topology I – Metric Spaces I.