# metric spaces: {\$ x \$} is open or closed in \$ (X, d) \$? Test the result

To demonstrate that $${x }$$ is closed, one can show that $$X setminus {x }$$ It's open.

Leave $$y in X setminus {x }$$ and establish $$varepsilon = d (x, y)$$ where $$d$$ It is your given metric. But $$x not in B (and, varepsilon)$$, Which means that $$B (y, epsilon) subset X setminus {x }$$. But that clearly gives $$X setminus {a }$$ is open, which implies that $${x }$$ The test is closed and completed.

Note: As Yanko pointed out in the comments, his given definition is incorrect.