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I have come up with an example when I trying to learn what first countability means
It says(from wikipedia)
In a metric space, for any point x, the sequence of open balls around x with radius 1/n form a countable neighbourhood basis [tex]\mathcal{B}(x) = \{ B_{1/n}(x) ; n \in \mathbb N^* \}.[/tex] This means every metric space is first-countable.
According to definition of nhd basis we can express every open set can be expressed as a union of elts of nhd basis But for example [tex] \{ B_{ sqrt{2} }(x) [/tex] can not be expressed in this way
Here is the question
which point did i miss?
It says(from wikipedia)
In a metric space, for any point x, the sequence of open balls around x with radius 1/n form a countable neighbourhood basis [tex]\mathcal{B}(x) = \{ B_{1/n}(x) ; n \in \mathbb N^* \}.[/tex] This means every metric space is first-countable.
According to definition of nhd basis we can express every open set can be expressed as a union of elts of nhd basis But for example [tex] \{ B_{ sqrt{2} }(x) [/tex] can not be expressed in this way
Here is the question
which point did i miss?