= Closed set
{title2=$X\setminus F\in\tau$}
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= Closed
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= Closed subset
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A <subset> $F$ of a <topological space> $(X,\tau)$ is <closed> when its <complement> in that same ambient space is <open>: $X\setminus F\in\tau$. For example, the <closed interval> $[0,1]$ is <closed> in the usual <real line> because its <complement> is the <open set> $(-\infty,0)\cup(1,\infty)$. Always specify the ambient <topological space>: a <subset> can be <closed> in a <subspace topology> without being <closed> in the surrounding space. https://www.maths.ox.ac.uk/system/files/attachments/Preliminary%20Reading%20%281%29.pdf[Richard Haydon, Oxford topology preliminaries, Section 0.1]{ref}
Being <closed> is not the logical negation of being <open>. In the usual <real line>, all four combinations occur: $(0,1)$ is <open> but not <closed>; $[0,1]$ is <closed> but not <open>; the half-open <interval> $[0,1)$ is neither <open> nor <closed>; and both the <empty set> and the whole <real line> are <open> and <closed>, hence <clopen>. Nor does being <closed> imply being a <bounded set>: $[0,\infty)$ is <closed> because its <complement> $(-\infty,0)$ is <open>, but it is not a <bounded set>. The symbol $\infty$ is not an extra point that this <interval> must contain.
\Image[/open-and-closed-sets-on-line.png]
{title=The four combinations of openness and closedness in the real line}
The coloured strokes mark membership; filled endpoint circles mean included points, and hollow endpoint circles mean excluded points. The <empty set> has no coloured points, while arrows at both ends of the coloured <real line> indicate that membership continues indefinitely. These examples use the usual <topology> of the <real line>.
In a <metric space> $X$, a <subset> $F$ is <closed> exactly when every <convergent sequence> $(x_n)$ in $F$ whose <limit of a sequence> $x$ lies in $X$ has $x\in F$. For example, $x_n=1-1/n$ lies in $[0,1)$ for $n\geq1$ and has <limit of a sequence> $1$, which is missing; thus this half-open <interval> is not <closed> in the <real line>. This criterion is useful for keeping limiting constructions inside a prescribed <set>. https://ocw.mit.edu/courses/18-100b-real-analysis-spring-2025/mit18_100b_s25_lec_full.pdf[MIT Real Analysis, Lecture 13, closed sets and convergent sequences]{ref}
In every <topological space>, the general criterion is $F=\overline F$, using <closure>; equivalently, $F$ contains all its <limit points>. A <closed set> is always a <sequentially closed set>, but the converse can fail outside <metric spaces>. Arbitrary <intersections> and finite <unions> of <closed sets> are <closed>, by taking <complements> in the <topology axioms>.
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