- E40
[0PS] Prerequisites:[0M7].Let \(X\) be a set with two distances \(d_ 1,d_ 2\); letβs call \(π_ 1,π_ 2\) respectively the induced topologies. We have that \(π_ 1β π_ 2\) if and only if
\[ β xβ X~ β r_ 1{\gt}0~ β r_ 2{\gt}0 ~ :~ B^ 2(x,r_ 2)β B^ 1(x,r_ 1) \]where
\[ B^ 2(x,r_ 2)=\{ yβ X:d^ 2(x,y){\lt}r_ 2\} \quad ,\quad B^ 1(x,r_ 1)=\{ yβ X:d^ 1(x,y){\lt}r_ 1\} \quad . \]Note that this exercise is the analogue in metric spaces of the principle [0M7] for the bases of topologies.
EDB β 0PS
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Authors:
"Mennucci , Andrea C. G."
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- accumulation point, in metric spaces
- topology, in metric spaces
- metric space
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