I'm wondering if a set is open/closed/compact/connected independent of the chosen metric. For example, if a set is compact, does that mean it's compact regardless of the metric defined? I think the answer is no but cannot come up with an example. Can anyone help me with it?
2026-04-03 20:33:21.1775248401
Is a set open/closed/compact/connected independent of the metric defined on it?
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This is true if and only if the two metrics induce the same topology (simply by definition), which need not be true. For example, you can define a metric $d$ on the real numbers with $d(x, y) =1$ if $x\neq y$. Then every subset is open, which is certainly not true in the usual topology.