## Getting acquainted with open and closed sets

The open sphere with center $$x$$, radius $$r$$ in $$\mathbb R^2$$, and the Eucledian metric is for the purpose of this problem denoted as $$B(x,r)$$.

Consider the metric space $$(X,\rho)$$, where $$X = \{(0, 0)\} \cup B((0, 2), 1)$$ is a subset of $$\mathbb R^2$$ and $$\rho$$ is the Euclidean metric restricted to the set $$X$$.

Determine whether the following subsets $$X$$ are open or closed or both.

• #### Variant 1

$$\{(x_1, x_2) \in X\colon x_1 = 0 \}$$.

• #### Variant 2

$$\{(0, 0)\}$$.

• #### Variant 3

$$B((0, 2), \frac12)$$.