"Exercise 1. Show that if $X$ is equipped with the discrete metric $d$ then every subset of $X$ is both open and closed. Deduce that any function $f : (X, d) → (Y, dY )$ is continuous."
My lecturer shared the following answer:
"Exercise 1. If $A\subset{X}$ then for every $a\in{A}$ we have $B(a, 1/2) = \{a\}$, and so $A$ is open. Since every subset is open, every subset is also closed. The function $f$ is continuous if $f^{-1}(U)$ is open in $(X,d)$ for any open set U in $(Y,dY )$; but $f^{-1}(U)$ is a subset of $(X,d)$, so is always open."
So my question is that are singletons open? I thought they are closed. Or does it depend on the metric?
Best Answer
In the case of discrete metric all sets are both open and closed. In particular singletons are open and closed. In general singletons in a metric space are closed. They need not be open as in the case of the real line with usual metric.