Elementary Properties of the Closed Ball in a Metric Space

Let $(X,d)$ be a metric space, let $x\in X$, let $r$ be a real number with $0\le r$, the order $\le$ being that of the ordered field of real numbers, and let $\bar{B}_d(x,r)$ be the closed ball. Let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$, a topology on $X$ by Metric Open Sets Form a Topology.

Then the following hold.

1. (Centre, and comparison with the open ball) $x\in\bar{B}_d(x,r)$; and if $0<r$, then the open ball satisfies $B_d(x,r)\subseteq\bar{B}_d(x,r)$.

2. (Boundedness) $\bar{B}_d(x,r)$ is bounded in $(X,d)$.

3. (Closedness) $\bar{B}_d(x,r)$ is a closed subset of the topological space $(X,\mathcal{T}_d)$.