Let $(X,d)$ be a metric space and $x_0∈X$ be a limit point of $X-{x_0}$

Question

Let $(X,d)$ be a metric space and $x_0∈X$ be a limit point of $X-{x_0}$. Direct the set $X-{x_0}$ by the relation $x ≤ x'$ if $d(x',x_ 0)≤ d(x,x_0)$. Show that a net $ϕ:X-{x_0} rightarrow Y$, where $Y$ is a metric space, converges to $y_0∈ Y$ if and only if $lim_{xto x_0} ϕ (x)=y_0$.
When I write the definition of the limit for metric spaces, I realize that the $delta$ controls the directed set and the net controls the $epsilon$, but I am not sure how to write it.

Brian M. Scott · Answer

I’ll do one direction to get you started.
Suppose that $varphi$ converges to $y_0$, and let $epsilon>0$. Then there is an $x_epsilonin Xsetminus{x_0}$ such that $d_Y(varphi(x),y_0)<epsilon$ whenever $xin Xsetminus{x_0}$ and $x_epsilonle x$, i.e., whenever $xin  Xsetminus{x_0}$ and $d_X(x,x_0)le d_X(x_epsilon,x_0)$. Take $delta=d_X(x_epsilon,x_0)$: if $d_X(x,x_0)<delta$, then $d_Y(varphi(x),y_0)<epsilon$, so $lim_limits{xto x_0}varphi(x)=y_0$.

Let $(X,d)$ be a metric space and $x_0∈X$ be a limit point of $X-{x_0}$

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