---- > [!definition] Definition. ([[dense]]) > A subset $D$ of a [[topological space]] $X$ is said to be **dense in $X$** if is [[closure]] $\overline{D}$ equals $X$. > [!equivalence] Equivalence for first-countable spaces. > The immediate corollary of [[the sequence lemma]] below helps characterize dense subsets, especially for [[first-countable space|first-countable spaces]]. > Suppose that for all $x \in X$ there exists a [[sequence]] $(d_{n})$ in $D$ converging to $x$. Then $D$ is dense in $X$. > The converse holds if $X$ is first-countable. That is, if $X$ is [[first-countable space|first-countable]], then $D \subset X$ is dense in $X$ if and only if for all $x \in X$, there exists a sequence $(d_{n})$ of elements in $D$ [[converge|converging]] to $x$. ^equivalence > [!equivalence] Equivalence for (pseudo)metric spaces. > Let $(X, d)$ be a [[metric space]] or [[pseudometric|pseudometric space]]. By [[neighborhood-basis characterization of set closure|this result]], $D \subset X$ is dense in $X$ if and only if for all $x \in X$ and $\varepsilon>0$ there exists $a \in D$ such that $d(x, a)<\varepsilon$. ^equivalence > [!basicexample] > - [[the rationals are dense in the reals]] > - [[on the circle, continuous functions are dense in Riemann integrable functions with respect to the 1-seminorm]] > - [[Fourier series is Cesaro summable at points of continuity|trigonometric polynomials are dense (uniformly) in the continuous functions on the circle]] ---- #### ---- #### 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 > ```