Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: *[[Specializations]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* Properties:: [[closed sets behave complementarily to open sets]], [[characterization of closed sets in subspaces]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* Examples:: [[topology from closed sets]] ---- > [!definition] Definition. ([[closed set]]) > A subset $A$ of a [[topological space]] $X$ is said to be **closed in $X$** if the set $X \cut A$ is [[open set|open in]] $X$. > [!equivalence] Equivalence for first-countable spaces. > If $X$ is [[first-countable space|first-countable]], then $A \subset X$ is [[closed set|closed]] if and only if $A$ is [[the sequence lemma|sequentially closed]], as a [[the sequence lemma|corollary]] of [[the sequence lemma]]. ^equivalence > [!basicexample] > 1. The subset $[a,b]$ of $\mathbb{R}$ is **closed in $\mathbb{R}$** because its complement $(-\infty, a) \cup (b, \infty)$ > is [[open set|open in]] $\mathbb{R}$ as a union of (arbitrarily many) [[open interval]]s. > 2. In $\mathbb{R}^{2}$ the set $\{ (x,y): x \geq 0 \and y \geq 0 \}$ > is [[closed set|closed]], because its complement $\{ x,y \}: x < 0 \text{ or } y < 0$ > is the union $(-\infty,0) \times \mathbb{R} \cup \mathbb{R} \times (-\infty,0)$ > is the product of [[open sets]] in $\mathbb{R}$; hence open in the [[product topology]]. > 3. In the [[finite complement topology|cofinite topology]] on a set $X$, the [[closed set]]s consist of $X$ itself and all finite subsets of it. > 4. In the [[discrete topology]] on a set $X$, every set is **closed**. > 5. Consider the following subset of $\mathbb{R}$: $Y=[0,1] \cup (2,3),$ > in the [[subspace topology]]. In this [[subspace topology|space]], the set $[0,1]$ is [[open set|open]] as the intersection of $\left( -0.1, 1.1 \right)$ with $Y$; similarly, $(2,3)$ is [[open set|open]] as a subset of $Y$. Since $[0,1]$ and $(2,3)$ are complements in $Y$ of each other, each is [[closed set|closed]] as subsets of $Y$ as well! ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` # Appendix: math 395 legacy ... general definition - see [[topological space]] # Definition Let $X$ be a [[metric space]]. A subset $U$ of $X$ is said to be *closed* in $X$ if its complement $X \backslash C$ is an [[open set]] in $X$. # Theorems - Let $(X,d)$ be a [[metric space]]. Then *finite intersections* and *arbitrary unions* of closed sets in $X$ are closed in $X$ (shown in [[math 297]], iirc). - Let $X$ be a [[metric space]]; let $Y$ be a metric subspace. A subset $A$ of $Y$ is closed in $Y$ iff it has the form $A = C \cap Y$, where $C$ is closed in $X$.