Is the Koch Snowflake a Compact Space? I am taking an introductory topology class, and we recently defined the notion of compactness. Earlier in the chapter, the Koch snowflake is described, and I am wondering: is the Koch snowflake a compact set? Intuitively I think the answer is yes: it is ~~an infinite union of closed sets~~ (not necessarily closed) and its limiting area is bounded, so it is a bounded and closed set of $\mathbb{R}^{2}$. Am I just waiving my hands, or is this a solid argument?

The union of infinitely many closed sets is not necessarily closed. However, the Koch snowflake is compact: it’s the range of a continuous function defined on the compact set $[0,1]$, and continuous functions preserve compactness.