Topological Properties closed sets

Question

I'm slightly confused with the idea of 'topological properties' Is closed-ness of a subset of a metric space X a topological property? I think it is because if a subset is closed under 1 metric then it should be closed under another?

Answer 1

You are right, but for the wrong reason. Being closed is a topological property because it depends only upon the topology induced by the metric, not by the metric itself. For instance, being bounded is not a topological property, because a set may be bounded with respect to a metric and unbounded with respect to another equivalent metric.

On the other hand, it is false that if a set is closed with respect to a metric then it will automatically be closed with respect to any other metric. For instance, in $\mathbb R$, $[0,1)$ is closed with respect to the discrete metric, but not with respect to the usual one.