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

Mathematics Asked by Calmat on October 1, 2020

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.

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 $$d_Y(varphi(x),y_0) whenever $$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), then $$d_Y(varphi(x),y_0), so $$lim_limits{xto x_0}varphi(x)=y_0$$.

Answered by Brian M. Scott on October 1, 2020

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