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Let $(X,\tau)$ be a topological space. Then a subset $C\subseteq X$ is closed if its complement $X\setminus C$ is open under the topology $\tau$.
Examples:

In any topological space $(X,\tau)$, the sets $X$ and $\varnothing$ are always closed.

Consider $\mathbb{R}$ with the standard topology. Then $[0,1]$ is closed since its complement $(\infty,0)\cup(1,\infty)$ is open (being the union of two open sets).

Consider $\mathbb{R}$ with the lower limit topology. Then $[0,1)$ is closed since its complement $(\infty,0)\cup[1,\infty)$ is open.
Closed subsets can also be characterized as follows:
A subset $C\subseteq X$ is closed if and only if $C$ contains all of its cluster points, that is, $C^{{\prime}}\subseteq C$.
So the set $\{1,1/2,1/3,1/4,\ldots\}$ is not closed under the standard topology on $\mathbb{R}$ since $0$ is a cluster point not contained in the set.
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