5. Let (X, d) be a metric space. i) Show that the open ball B, (xo) = {x € X : d (x, xo)


5. Let (X, d) be a metric space.<br>i) Show that the open ball B, (xo) = {x € X : d (x, xo) < r} is an open set in (X, d).<br>ii) Show that the closed ball C, (xo)={x € X : d (x, xo) <r} is a closed set in (X, d).<br>iii) Show that cl (B, (xo)) C C, (xo). State any property that you use. Show that this<br>inclusion can be strict if d is the discrete metric.<br>

Extracted text: 5. Let (X, d) be a metric space. i) Show that the open ball B, (xo) = {x € X : d (x, xo) < r}="" is="" an="" open="" set="" in="" (x,="" d).="" ii)="" show="" that="" the="" closed="" ball="" c,="" (xo)="{x" €="" x="" :="" d="" (x,="" xo)="">

Jun 04, 2022
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