Neighborhood

Let (X,d)(X,d)(X,d) be a metric space and let x∈Xx\in Xx∈X. A set N⊆XN\subseteq XN⊆X is a neighborhood of xxx if there exists r>0r>0r>0 such that B(x,r)⊆NB(x,r)\subseteq NB(x,r)⊆N (see open ball ). Neighborhoods encode the local structure around a point. Many definitions in analysis can be phrased using neighborhoods (e.g., limit points , closure , continuity ). ...

Nested Interval Theorem

Nested Interval Theorem: Let In=[an,bn]I_n=[a_n,b_n]In=[an,bn] be closed intervals in R\mathbb{R}R such that In+1⊆Infor all n,I_{n+1}\subseteq I_n \quad \text{for all } n,In+1⊆Infor all n, and lim⁡n→∞(bn−an)=0\lim_{n\to\infty}(b_n-a_n)=0limn→∞(bn−an)=0. Then ⋂n=1∞In={x}\bigcap_{n=1}^\infty I_n = \{x\}⋂n=1∞In={x} for a unique x∈Rx\in\mathbb{R}x∈R. ...

Newton–Leibniz formula

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be continuous and let F:[a,b]→RF:[a,b]\to\mathbb{R}F:[a,b]→R satisfy F′(x)=f(x)F'(x)=f(x)F′(x)=f(x) for all x∈(a,b)x\in(a,b)x∈(a,b). Corollary (Newton–Leibniz): ∫abf(x) dx=F(b)−F(a). \int_a^b f(x)\,dx = F(b)-F(a). ∫abf(x)dx=F(b)−F(a). This is the standard evaluation rule for definite integrals using antiderivatives . ...

Nowhere dense set

Let (X,d)(X,d)(X,d) be a metric space and let A⊆XA\subseteq XA⊆X. The set AAA is nowhere dense in XXX if int⁡(A‾)=∅, \operatorname{int}(\overline{A})=\varnothing, int(A)=∅, where A‾\overline{A}A is the closure and int⁡\operatorname{int}int denotes the interior . Equivalently, AAA is nowhere dense iff every nonempty open set U⊆XU\subseteq XU⊆X contains a nonempty open set V⊆UV\subseteq UV⊆U with V∩A=∅V\cap A=\varnothingV∩A=∅. Nowhere dense sets are “small” from the point of view of category (not measure). ...

One-sided limit

Let f:E→Rf:E\to\mathbb{R}f:E→R (or C\mathbb{C}C) with E⊆RE\subseteq\mathbb{R}E⊆R, and let a∈Ra\in\mathbb{R}a∈R be a limit point of E∩(a,∞)E\cap(a,\infty)E∩(a,∞). The right-hand limit of fff at aaa is LLL (written lim⁡x→a+f(x)=L\lim_{x\to a^+} f(x)=Llimx→a+f(x)=L) if ∀ε>0, ∃δ>0 such that ∀x∈E, (0<x−a<δ⇒∣f(x)−L∣<ε).\forall \varepsilon>0,\ \exists \delta>0\ \text{such that}\ \forall x\in E,\ \bigl(0<x-a<\delta \Rightarrow |f(x)-L|<\varepsilon\bigr).∀ε>0, ∃δ>0 such that ∀x∈E, (0<x−a<δ⇒∣f(x)−L∣<ε). Similarly, the left-hand limit lim⁡x→a−f(x)=L\lim_{x\to a^-} f(x)=Llimx→a−f(x)=L uses the condition 0<a−x<δ0<a-x<\delta0<a−x<δ (equivalently, −δ<x−a<0-\delta<x-a<0−δ<x−a<0). ...

Open ball

Let (X,d)(X,d)(X,d) be a metric space , let x∈Xx\in Xx∈X, and let r>0r>0r>0. The open ball of radius rrr centered at xxx is B(x,r):={y∈X:d(x,y)<r}.B(x,r):=\{y\in X : d(x,y)<r\}.B(x,r):={y∈X:d(x,y)<r}.Open balls are the basic building blocks of the topology induced by a metric : open sets are exactly those that contain an open ball around each of their points. ...

Open set

Let (X,d)(X,d)(X,d) be a metric space . A subset U⊆XU\subseteq XU⊆X is open if for every x∈Ux\in Ux∈U there exists r>0r>0r>0 such that B(x,r)⊆U.B(x,r)\subseteq U.B(x,r)⊆U.Open sets are the primitive “admissible neighborhoods ” in topology. In analysis, openness is the natural condition for local arguments (e.g., differentiability is typically defined on open subsets of Rk\mathbb{R}^kRk). ...

Open sets form a topology

Open sets form a topology: Let (X,d)(X,d)(X,d) be a metric space . Then: ∅\varnothing∅ and XXX are open ; arbitrary unions of open sets are open: if {Uα}α∈A\{U_\alpha\}_{\alpha\in A}{Uα}α∈A are open, then ⋃α∈AUα\bigcup_{\alpha\in A} U_\alpha⋃α∈AUα is open; finite intersections of open sets are open: if U1,…,UnU_1,\dots,U_nU1,…,Un are open, then ⋂j=1nUj\bigcap_{j=1}^n U_j⋂j=1nUj is open. These closure properties justify treating “open sets” as the primitive objects defining the topological structure induced by a metric . ...

Operator norm

Let (V,∥⋅∥V)(V,\|\cdot\|_V)(V,∥⋅∥V) and (W,∥⋅∥W)(W,\|\cdot\|_W)(W,∥⋅∥W) be normed vector spaces over F\mathbb{F}F, and let T:V→WT:V\to WT:V→W be linear. The operator norm (or induced norm) of TTT is ∥T∥:=sup⁡{∥Tx∥W:x∈V, ∥x∥V=1}. \|T\| := \sup\{\|T x\|_W : x\in V,\ \|x\|_V=1\}. ∥T∥:=sup{∥Tx∥W:x∈V, ∥x∥V=1}. Equivalently, ...

Order axioms (for R as an ordered field)

The order axioms for R\mathbb{R}R assert that there is a total order ≤\le≤ such that: (Trichotomy) for all a,b∈Ra,b\in\mathbb{R}a,b∈R, exactly one of a<ba<ba<b, a=ba=ba=b, a>ba>ba>b holds, (Transitivity) a≤ba\le ba≤b and b≤cb\le cb≤c imply a≤ca\le ca≤c, (Compatibility with addition) if a≤ba\le ba≤b then a+c≤b+ca+c\le b+ca+c≤b+c for all ccc, (Compatibility with multiplication) if 0≤a0\le a0≤a and 0≤b0\le b0≤b then 0≤ab0\le ab0≤ab. Together with the field axioms , these make R\mathbb{R}R an ordered field, enabling inequalities, monotonicity arguments, and notions like boundedness and supremum . ...