Getting acquainted with open and closed sets
Task number: 3165
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
\(B((0, 2), \frac12)\).
