Hi. Can you explain how the union of arbitrary open nballs forms a topology? It would seem to fail the requirement that any intersection of open sets is itself an open set. The intersection of an nball with another nball isn't an nball.
They form a base/basis for the topology induced by a metric. Any set U that would be considered open can be written as a union of the nballs contained within S. Right the intersection of nballs is not necessarily an nball, but that’s not required. The intersection of two open sets U1 and U2 will be open because for any point x you pick in the intersection, you can find an nball centered at x that’s contained in the union.
Amazing, thank you man (:
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Hi. Can you explain how the union of arbitrary open nballs forms a topology? It would seem to fail the requirement that any intersection of open sets is itself an open set. The intersection of an nball with another nball isn't an nball.
They form a base/basis for the topology induced by a metric. Any set U that would be considered open can be written as a union of the nballs contained within S. Right the intersection of nballs is not necessarily an nball, but that’s not required. The intersection of two open sets U1 and U2 will be open because for any point x you pick in the intersection, you can find an nball centered at x that’s contained in the union.