Consider a Metric Space(X,Td,d)U∈Td⟺∀y∈U,∃δ>0s.t.Bd(y,δ)⊂U
Facts
Every point of U∈T is an interior point of U.
Consider a Topological Space(X,T) and a subset U⊂X. Then,
U is open in X if and only if for an arbitrary point x in U, there exists an open subset Ux⊂U containing x.
U∈T⟺∀x∈U,∃Ux∋xs.t.Ux∈TandUx⊂U
A is an open set if and only if it’s equal to its InteriorA=intA.
Consider a set X with two topologiesT1 and T2.
If T1⊂T2, then T1 is weaker (smaller or coarser) than T2 and T2 is stronger (larger or finer) than T1.
Comparison of Topologies with Basis
Consider two topological spaces with bases(X,T,B) and (X,T′,B′). Then,
T⊂T′⟺(∀x∈X,∀B∈Bs.t.x∈B),∃B′⊂B′s.t.B′⊂B
A basis B is a collection of subsets of X such that
∀x∈X,∃B∈Bs.t.x∈BorBi∈B⋃Bi=X
∀B1,B2∈B,∀x∈B1∩B2,∃B3s.t.B3⊂B1∩B2
B⊂Ts.t.(∀U∈T,∃{Bi}i∈I⊂Bs.t.U=⋃i∈IBi)
Or, equivalently, a basis B for the TopologyT of a Topological Space(X,T) is a family of open subsets of X such that every Open Set can be expressed as a union of subfamily of the family B.
Examples
A Standard Topology on real numbers(R,U) has a countable basis B={(a,b)⊂R∣a,b∈Q}
A lower limit topology Tl generated by Bl={[a,b)⊂R∣a<b} does not have a countable basis.
Facts
A basis can be constructed for a given Topology, where the basis is not unique.
B⊂Ts.t.(∀U∈T,∀x∈U,∃B∈Bs.t.x∈B⊂U)
Suppose that a Topological Space(X,T) has a countable basis B={Bn}n∈N. Then,
Consider a Topological Space(X,T) has a countable basis B={Bn}n∈N, and a Subspace Topology(Y,TY) where Y⊂X. Then,
The subspace topology also has a countable basis.
Consider a set X and a family of functions {fi:X→Xi∣i∈I} from X to each Topological Space(Xi,Ti).
The weak topology generated by the family of functions fi is the Topology determined by the Subbasis of open sets{fi−1(Ui)∣i∈I,Ui∈Ti}.
Examples
Consider a metric space (X,d), a point a∈X, and a function f:X→R0+,f(x)=d(a,x).
Then the weak topology generated by the function f is a Metric Topology determined by a Subbasisf−1([0,r))=B(a,r)). It is an open ball with center a and radius r.
Consider a set Rω, and a family of projection functions{πi:Rω→(R,U)}.
The weak topology generated by the family of functions is a Product Topology determined by a Subbasisπi−1(U) where U∈U.
Facts
Consider a Topological Space(X,T) and a family of continuous functionsF={fi:X→Yi} with domain X.
Then, the weak topology generated by the family of functions F is weaker than T.
A subbasis S for a topology T on X is a collection of subsets of X whose union equals X.
S⊂2Xs.t.⋃S=X
Or, Equivalently, a subset S of T is a subbasis for T if the family of all finite intersections of members of the collection {⋂u=1nSi∣Si∈S,n∈N} is a basis for T.
Basis Generated By a Subbasis
B={⋂i=1nSi∣Si∈S}
A basis can be defined as a set of all finite intersections of a subbasis.
Examples
A Standard Topology on R can be generated by a subbasis S={(a,∞),(−∞,b)⊂R∣a,b∈R}.
Every basis element B∈B={(a,b)⊂R∣a<b} can be generated by the subbasis S.
An order topology is a specific topology that can be defined on any Totally Ordered Set.
A basis of the topology is defined as:
B={(a,b),[a0,b),(a,b0]⊂X∣a<b}
where a0 and b0 are the smallest and the largest element (if any) respectively.
Examples
R2 with dictionary order: B={((a,b),(c,d))⊂R2∣a<cor(a=candb<d)}
A product topology is the Cartesian product of a family of topological spaces
Let i∈I be an index set, and Xi be a Topological Space. The product topology is defined as
X:=∏i∈IXi={x:I→⋃i∈IXi∀i∈I,xi∈Xi}
where xi is the i-th coordinate (component) of the function x.
Facts
Consider bases BX,BY for the topologies on X,Y. Then, BX×Y={BX×BY∣BX∈BY,BY∈BY} is a basis for the topology on X×Y.
Given a topological space (X,T) with a basis B, and a subset of the set Y⊂X.
The subspace topology on Y and its basis is defined as
TY={U∩Y∣U∈T},BY={B∩Y∣B∈B}
Consider a subset A of a Topological Space(X,T). The interior of A is defined as
\mathring{A} = \operatorname{int}A
&:= \{ x \in X | \exists U \in \mathcal{T}\ \text{s.t.}\ x\in U \subset A \} \\
&= \bigcup \{ U | U \in \mathcal{T}\ \text{and}\ U \subset A \} \\
\end{aligned}$$
The interior of the $A$ can be defined in any of the following equivalent ways:
- $\operatorname{int}A$ is the set of all interior points of $A$
- $\operatorname{int}A$ is the largest open subset of $X$ contained in $A$
- $\operatorname{int}A$ is the union of all open sets of $X$ contained in $A$
## Definition for Metric Space
$$\mathring{A} = \operatorname{int}A = \{x | \exists\epsilon>0, B(x, \epsilon) \subset A\}$$
where $B(x, \epsilon)$ is an open ball with a center $x$ and radius $\epsilon$.
# Facts
> For subsets $A, B$ of a [[Topological Space]] $(X, \mathcal{T})$,
> $$A \subset B \Longrightarrow \operatorname{int} A \subset \operatorname{int} B$$
> For subsets $A, B$ of a [[Topological Space]] $(X, \mathcal{T})$,
> $$\operatorname{int}(A \cap B) = \operatorname{int} A \cap \operatorname{int} B$$Link to original
Closure
Definition
Consider a subset A of a Topological Space(X,T). The closure of A is defined as
\bar{A} = \operatorname{cl}A
:=& \{x \in X | \forall \mathcal{N}_{x},\ \mathcal{N}_{x} \cap A \neq \emptyset\}\\
=& \bigcap \{ C | (X \setminus C) \in \mathcal{T}\ \text{and}\ A \subset C\}\\
=& A \cup A'\\
=& A \cup \partial A
\end{aligned}$$
where $\mathcal{N}_{x}$ is a [[Neighborhood]] of $x$.
The closure of the $A$ can be defined in any of the following equivalent ways:
- $\operatorname{cl}A$ is the set of all points of closure of $A$.
- $\operatorname{cl}A$ is the intersection of all [[Closed Set|closed sets]] containing $A$.
- $\operatorname{cl}A$ is the union of $A$ and the set of all its [[Limit Point|limit points]] $A'$.
- $\operatorname{cl}A$ is the union of $A$ and its [[Boundary]] $\partial A$
# Facts
> ![[Pasted image 20241115121613.png|300]]
>
> Consider a subspace topology $(Y, \mathcal{T}_{Y})$ of a [[Topological Space]] $(X, \mathcal{T})$ and a subset $A \subset Y$. Then,
> $$\operatorname{cl}_{Y} A = \operatorname{cl}_{X} A \cap Y$$
> For subsets $A, B$ of a [[Topological Space]] $(X, \mathcal{T})$,
> $$A \subset B \Longrightarrow \bar{A} \subset \bar{B}$$
> For subsets $A, B$ of a [[Topological Space]] $(X, \mathcal{T})$,
> $$\overline{A \cup B} = \bar{A} \cup \bar{B}$$Link to original
Boundary
Definition
Consider a subset A of a Topological Space(X,T). The boundary of A is defined as
\partial A = \operatorname{bd}A
&:= \bar{A} \setminus \operatorname{int}A \\
&= \bar{A} \cap \overline{(X \setminus A)} \\
&= X \setminus [\operatorname{int} A \cup \operatorname{int}(X\setminus A)]
\end{aligned}$$
The boundary of the $A$ can be defined in any of the following equivalent ways:
- $\partial A$ is the [[Closure]] of $A$ minus the [[Interior]] of $A$ in $X$.
- $\partial A$ is the intersection of the [[Closure]] of $A$ with the [[Closure]] of its complement.Link to original
Neighborhood
Definition
Consider a subset A of a Topological Space(X,T).
A neighborhood Nx of a point x∈X is a subset of X including an open set U∈T containing x.
x∈U⊂Nx⊂T
Consider a subset A of a Topological Space(X,T).
A point x∈X is a limit point of A if every Neighborhood of x contains a point of A different from x itself.
x∈Xs.t.∀Nx,Nx∩(A−{x})=∅
And the set of all limit points of A is notated as A′A′={x∈X∣∀Nx,Nx∩(A−{x})=∅}
The separation axioms are properties of topological spaces that characterize how distinct points and closed sets can be separated by open sets. These axioms form a hierarchy of increasingly stronger conditions:
Consider two topological spaces(X,TX,BX) and (Y,TY,BY). A function f:X→Y is continuous if
\forall V \in \mathcal{T}_{Y},\ f^{-1}(V) \in \mathcal{T}_{X}\
&\Longleftrightarrow\ \forall B \in \mathcal{B}_{Y},\ f^{-1}(B) \in \mathcal{T}_{X}\\
&\Longleftrightarrow\ \forall A \subset X,\ f(\bar{A}) \subset \overline{f(A)}\\
&\Longleftrightarrow\ \forall (X\setminus B) \in \mathcal{T}_{Y},\ (X\setminus f^{-1}(B)) \in \mathcal{T}_{X}\\
&\Longleftrightarrow\ \forall x \in X,\ \exists \mathcal{N}_{x}\ \text{s.t.}\ f(\mathcal{N}_{x}) \subset \mathcal{N}_{f(x)}
\end{aligned}$$
where $\mathcal{N}_{x}$ is a [[Neighborhood]] of $x$.
A function is continuous if and only if the inverse image of any arbitrary open set in [[Codomain]] is an open set.
## Continuity of Real-Valued Function
### Continuous at a Point
Suppose $f: \mathcal{D} \subset \mathbb{R}^{n} \to \mathbb{R}^{m}$, and $\mathbf{x}_{0} \in \mathcal{D}$
$$\begin{aligned}
\lim\limits_{\mathbf{x} \to \mathbf{x}_{0}} f(\mathbf{x}) = f(\mathbf{x}_{0}) &\Leftrightarrow \forall \epsilon > 0, \exists \delta > 0, \forall x \in \mathcal{D}, (|\mathbf{x}-\mathbf{x}_{0}| < \delta \Rightarrow |f(\mathbf{x})-f(\mathbf{x}_{0})| < \epsilon)\\
&\Leftrightarrow \forall\epsilon>0, \exists\delta>0\ \text{s.t}\ f(B_{\delta}(\mathbf{x})) \subset B_{\epsilon}(f(\mathbf{x}))\\
&\Leftrightarrow \forall\epsilon>0, \exists\delta>0\ \text{s.t}\ B_{\delta}(\mathbf{x}) \subset f^{-1}(B_{\epsilon}(f(\mathbf{x})))
\end{aligned}$$
$f$ is continuous at a point $\mathbf{x}_{0}$ if the [[Limit of a Function|limit]] of $f(\mathbf{x})$, as $\mathbf{x}$ approaches $\mathbf{x}_{0}$, exists and is equal to $f(\mathbf{x}_{0})$
### Continuous on an Open Interval
Suppose $f: \mathcal{D} \subset \mathbb{R}^{n} \to \mathbb{R}^{m}$, and $X \subset \mathcal{D}$
$$\forall \mathbf{x}_{0} \in X, \lim\limits_{\mathbf{x} \to \mathbf{x}_{0}} f(\mathbf{x}) = f(\mathbf{x}_{0})$$
A function is continuous at every point in an open interval $X$
### Continuous Function
Suppose $f: \mathcal{D} \subset \mathbb{R}^{n} \to \mathbb{R}^{m}$
$$\forall \mathbf{x}_{0} \in \mathcal{D},\ \lim\limits_{\mathbf{x} \to \mathbf{x}_{0}} f(\mathbf{x}) = f(\mathbf{x}_{0})$$
A function is continuous at every point in its domain
### Right-Continuous
Suppose $f: \mathcal{D} \to \mathbb{R}$, and $a \in \mathcal{D}$
$$\forall \epsilon > 0, \exists \delta > 0, \forall x \in \mathcal{D}, (0 \leq x-a < \delta \Rightarrow |f(x)-f(a)| < \epsilon) \Leftrightarrow \lim\limits_{x \to a^{+}} f(x) = f(a)$$
$f$ is right-continuous at $a$
The [[Limit of a Function#right-sided-limit|right-sided limit]] of $f(x)$, as $x$ approaches $a$ from the right side, exists and is equal to $f(a)$
### Left-Continuous
Suppose $f: \mathcal{D} \to \mathbb{R}$, and $a \in \mathcal{D}$
$$\forall \epsilon > 0, \exists \delta > 0, \forall x \in \mathcal{D}, (0 \leq a-x < \delta \Rightarrow |f(x)-f(a)| < \epsilon) \Leftrightarrow \lim\limits_{x \to a^{-}} f(x) = f(a)$$
$f$ is left-continuous at $a$
The [[Limit of a Function#left-sided-limit|left-sided limit]] of $f(x)$, as $x$ approaches $a$ from the left side, exists and is equal to $f(a)$
# Properties
## Construction of Continuous Functions
Suppose $a \in \mathcal{D}$, and $f, g: \mathcal{D} \to \mathbb{R}$ is continuous at $a$
Then, the following are satisfied
- $f + g$ is continuous at $a$
- $f - g$ is continuous at $a$
- $f \cdot g$ is continuous at $a$
- $g(a) \neq 0 \Rightarrow \cfrac{f}{g}$ is continuous at $a$
# Facts
![[Lipschitz Continuity#^e0a56a|^e0a56a]]
> Every continuous function $f: [a, b] \to \mathbb{R}$ is [[Darboux Integral|integrable]]
^c92817
![[Heine-Cantor Theorem]]
> Given two continuous functions $f: D_{f} \subset \mathbb{R}^{n} \to R_{f} \subset D_{g}$ and $g: D_{g} \subset \mathbb{R}^{m} \to R_{g} \subset \mathbb{R}^{l}$, then their composition $h := g \circ f: D_{f} \subset \mathbb{R}^{n} \to R_{g} \subset \mathbb{R}^{l}$ is continuous.
> A [[Constant Function]] $f: X \to Y,\quad f(x)=y_{0}$ is continuous.
> [[Inclusion Function|Inclusion Map]] $j: A \hookrightarrow X,\quad j(x) = x$ where $A \subset X$, is continuous
> $(f: X \to Y)\in C^{0}, (g: Y \to Z)\in C^{0}\ \Rightarrow\ (g\circ f: X \to Z) \in C^{0}$
> A [[Function Composition|composite function]] of continuous functions is continuous
> $(f: X \to Y) \in C^{0}, A \subset X\ \Rightarrow\ (f|_{A}: A \to Y) \in C^{0}$
> A continuous function with restricted [[Domain]] is continuous
> $(f: X \to Y) \in C^{0}, f(x) \subset Z \subset Y\ \Rightarrow\ (g: X \to Z,\quad g(x)=f(x)) \in C^{0}$
> $(f: X \to Y) \in C^{0}, Y \subset Z\ \Rightarrow\ (h: X \to Z,\quad h(x)=f(x)) \in C^{0}$
> A continuous function with expanded and restricted [[Codomain]] is continuous.
> $(f|_{U_\alpha}: U_{\alpha}\to Y\ \text{s.t.}\ \bigcup_{\alpha} U_{\alpha} = X) \in C^{0} \Rightarrow (f: X \to Y) \in C^{0}$
> Consider a collection $(U_\alpha)$ of [[Open Set|open sets]] in $X$. If $X = \bigcup_{\alpha} U_{\alpha}= X$ and $f|_{U_\alpha}$ is continuous, then $f: X \to Y$ is continuous.
![[Gluing Lemma]]
> Consider [[Topological Space|topological Spaces]] $(A, \mathcal{T}), (X \times Y, \mathcal{T})$ and a function $f: A \to X\times Y,\quad f(a) = (f_{1}(a), f_{2}(a))$.
> $f \in C^{0} \Leftrightarrow (f_{1}: A \to X), (f_{2}: A \to Y) \in C^{0}$
> A function with a [[Product Topology]] [[Codomain]] is continuous if and only if all of its coordinate functions are continuous.
> Consider [[Topological Space|topological Spaces]] $(A, \mathcal{T}), (\prod\limits_{i \in \mathbb{N}}X_{i}, \mathcal{T})$ and a function $f: A \to \prod\limits_{i \in \mathbb{N}}X_{i},\quad f(a) = (f_{1}(a), f_{2}(a), \dots, f_{n}(a), \dots)$.
> $f \in C^{0} \Leftrightarrow \forall i \in \mathbb{N},\ (f_{i}: A \to X_{i}), \in C^{0}$
> A function with a countably infinite [[Product Topology]] [[Codomain]] is continuous if and only if all of its coordinate functions are continuous.
![[Closed Map#^b9de02|^b9de02]]Link to original
Gluing Lemma
Definition
Consider topological Spaces(X,T),(Y,TY), subsets A,B⊂Xs.t.(X∖A),(X∖B)∈TandX=A∪B, and functions (f:A→Y),(g:B→Y). Then,
f,g∈C0and(∀x∈A∩B,f(x)=g(x))⟹(h:X→Y,h(x)={f(x)g(x)x∈Ax∈B)∈C0
Two continuous functions whose domains are closed, and with overlapping areas having the same values can be glued together to form a Continuous Function.
A metric topology is a Topology generated by a basis defined by a Metricd.
Bd={Bd(x,ϵ)∣x∈X,ϵ>0}
where Bd(x,ϵ)={y∈X∣d(x,y)<ϵ} is an ϵ-ball centered at x.
Metrizable Topology
A Topological Space(X,T) is metrizable If there exist a Metricd on X that induces the topology T.
Facts
Consider two metric topologies (X,T,d),(X,T′,d′).
T⊂T′⟺∀x∈X,ϵ>0,∃δ>0s.t.Bd′(x,δ)⊂Bd(x,ϵ)
n→∞limfn=f uniformly
Suppose f:D→R, is a function, and (fn) is a Sequence of Functions whose term has the same domain as the function f
If ∀ϵ>0,∃N∈N,∀x∈D,∀n≥N,∣fn(x)−f(x)∣<ϵ, then (fn) converges uniformly to f on D
Consider two quotient maps p:X→Y and f:Y→Z.
The composite map of the quotient maps f∘p is a quotient map
Consider a quotient map p:(X,TX)→(Y,TY) between two topological spaces, a subset A⊂Xs.t.p−1(p(A)), and a function p∣A:A→p(A) with restricted domain. Then,
If A is either open or closed in X, then the function p∣A is a quotient map
A∈TXor(X∖A)∈TX⟹p∣Ais a quotient map
If p is either an Open Map or a Closed Map, then the function p∣A is a quotient map.
Consider a quotient map p:(X,TX)→(Y,TY), and a function g:(X,TX)→(Z,TZ) satisfying ∀y∈Y,g∣p−1({y})=c where c is a constant, between topological spaces. Then,
There exists a function such that the composite functionf∘p is equal to g∃f:(Y,TY)→(Z,TZ)s.t.f∘p=g
A function f:(X,TX)→(Y,TY) between two topological spaces is a closed map if the function value of a closed set is a closed set.
(X∖C)∈TX⟹(Y∖f(U))∈TY
Consider a Topological Space(X,T) and a Surjective map p:X→A.
The quotient topology Tq on A consists of all sets U⊂A whose pre-image p−1(U) is Open Set in (X,T).
U∈Tq⟺p−1(U)∈T
Consider a Standard Topology on Real Number(R,T) and a quotient map p:R→{a,b,c},p(x)=⎩⎨⎧abcx>0x<0x=0.
The quotient topology Tq on the set {a,b,c} induced by p is generated as Tq={∅,a,b,{a,b},{a,b,c}}
Consider a Standard Topology on real plane (R2,T) and a unit disk X=D2={(x,y)∣x2+y2≤1} on it.
The Quotient Topology constructed by a partition X∗={(x,y)∣x2+y2<1}∪{S1} where S1={(x,y)∣x2+y2=1}, forms a sphere.
Consider a Standard Topology on real plane (R2,T) and a rectangle X={(x,y)∣0≤x≤1,0≤y≤1} on it.
Consider a partition X∗={(x,y)∣0<x<1,0<y<1}∪{(x,0),(x,1)∣0<x<1}∪{(0,y),(1,y)∣0<y<1}∪{(0,0)=(0,1)=(1,0)=(1,1)}
The Quotient Topology constructed by the partition forms a torus.
A surface is orientable if a normal vector can be consistently defined at every point of the surface. In other words, a coherent notation of up and down or inside and outside can be assigned across the entire surface. If a surface is not orientable, it is a non-orientable surface.
A Topological Space(X,T) is connected if it can not be represented as the union of two disjoint, non-empty, open or closed subsets.
∄A,B∈Ts.t.A,B=∅andA⊎B=X⟺∄(X∖A),(X∖B)∈Ts.t.A,B=∅andA⊎B=X
where ⊎ is a disjoint union.
Or, equivalently, if ∅,X are the only open and closed subsets.
Consider a Subspace Topology(Y,TY) of a Topological Space(X,T).
Then, A,B∈TYs.t.A,B=∅andA⊎B=Y is a separation of Y if and only if A⊎B=YandA′∩B=A∩B′=∅.
where ⊎ is a disjoint union, A′ is the set of all limit points of A.
Consider a Subspace Topology(Y,TY) of a Topological Space(X,T) and a separation {C,D} of X.
If (Y,TY) is a connected subspace, then Y⊂CorY⊂D.
Consider a collection of connected subspaces {Ui}i∈Is.t.Ui⊂X of a Topological Space(X,T).
If the subspaces have a point in common ∃x∈X,s.t.x∈i∈I⋂Ui⇔i∈I⋂Ui=∅, then the union of the collection i∈I⋃Ui is connected.
Consider a collection of connected subspaces{Ui}i∈Is.t.Ui⊂X of a Topological Space(X,T), and a connected subspace B.
If ∀i∈I,Ui∩B=∅, then B∪(i∈I⋃Ui) is connected.
Consider a Topological Space(X,T).
The connected component Cx of a point x∈X is the union of all connected subsets of X that contain x. It is the unique largest (with respect to ⊆) connected subset of X that contains x.
Facts
Each point x∈X is contained in exactly one component.
Given points x,y∈X, their components Cx,Cy are the same or disjoint.
∀x,y∈X,Cx=CyorCx∩Cy=∅
Every connected subset in X is contained in some component.
A Topological Space(X,T) is path connected if every pair of points of X can be joined by a path f:[0,1]→X in X where a path is a continuous map such that f(0)=x and f(1)=y.
∀x,y∈X,∃f:[0,1]→Xs.t.f∈C0,f(0)=x,f(1)=y
Consider a Topological Space(X,T).
The path component Px of a point x∈X is the union of all path connected subsets of X that contains x. It is the unique largest (with respect to ⊆) path connected subset of X that contains x.
Facts
Each point x∈X is contained in exactly one path component.
Given points x,y∈X, their path components Px,Py are the same or disjoint.
∀x,y∈X,Px=PyorPx∩Py=∅
Every path connected subset in X is contained in some path component.
Consider a Topological Space(X,T) and paths on it p1,p2 such that p1(1)=p2(0).
The path product of p1 and p2 is a path defined as
p1∗p2(t)=⎩⎨⎧p1(2t)p2(2t−1)0≤t≤2121≤t≤1
Consider a Topological Space(X,T)
The Topological Space is locally connected If every point in X is locally connected.
∀p∈X,∀U∈Ts.t.U∋p,∃connectedC∈Ts.t.p∈C⊂U
A Topological Space(X,T) is locally connected if and only if for each open subset U of the space X, each componentCU of the subset U is an Open Set.
∀U∈T,CU∈T
Consider a Topological Space(X,T)
The Topological Space is locally path connected If every point in X is locally path connected.
∀p∈X,∀U∈Ts.t.U∋p,∃path connectedC∈Ts.t.p∈C⊂U
A set is bounded if all of its points are within a certain distance of each other. The notion of boundedness is makes sense with some metric.
Consider an Partially Ordered SetX and a subset Y⊂X.
The set Y is bounded if it has both upper and lower bounds∃x0,x1∈X,s.t.(∀y∈Y,y≤x0)and(∀y∈Y,y≥x1)
where x0,x1 is the upper and lower bound of the set Y respectively.
Definition of Bounded Set on a Metric Space
Consider a Metric Space(X,T,d) and a subset Y⊂X.
The set Y is bounded if
∃r>0s.t.∀y0,y1∈Y,d(y0,y1)<r
Consider a Topological Space(X,T).
The space is compact if every open covering (a collection of open subsets {Ui}i∈I of X whose union is X) of X has a finite subcovering.
∀{Ui}i∈I⊂Ts.t.⋃i∈IUi=X,∃J⊂Is.t.∣J∣<∞and⋃j∈JUj=X
Consider a Subspace Topology(Y,TY) of a Topological Space(X,T). Then,
Y is compact if and only if every covering of Y by open sets in X has a finite subcollection covering Y.
∀{Ui}i∈I⊂TYs.t.⋃i∈IUi=Y,∃J⊂Is.t.∣J∣<∞and⋃j∈JUj=Y⟺∀{Ui}i∈I⊂Ts.t.⋃i∈IUi=Y,∃J⊂Is.t.∣J∣<∞and⋃j∈JUj=Y
Consider a Topological Space(X,T).
X is compact if and only if for every collection {Ci}i∈Is.t.(X∖Ci)∈T of closed sets in X that have the Finite Intersection Property, its entire intersection is a non-empty set i∈I⋂Ci=∅.
Consider a Hausdorff Space(X,T) and disjoint Compact subsets A,B⊂Xs.t.A∩B=∅.
Then, there exists disjoint open setsU,V∈Ts.t.U∩V=∅ satisfying A⊂U and B⊂V.
Consider a Topological Space(X,T) and a nested sequence (En)n∈Ns.t.∀n∈N,En+1⊂En of non-empty Compactclosed subsets of X.
Then, n∈N⋂En=∅
Nested Intervals Theorem
Consider a Sequence(In)n∈N, where In=[an,bn], of closed intervals.
The sequence of intervals (In)n∈N is called a sequence of nested intervals if
∀n∈N,In+1⊂Inandn→∞lim(bn−an)=0
If (In)n∈N is a sequence of nested intervals in real numbers, then
∃!x∈R,x∈n∈N⋂In
It is the special case of the Cantor’s intersection theorem.
Consider a Topological Space(X,T) and a collection of subsets {Ui}i∈I⊂X.
The collection has the finite intersection property if for every finite subcollection of {Ui}i∈I, the intersection is nonempty
{Ui}i∈I⊂Xs.t.(∀J⊂Is.t.∣J∣<∞,⋂j∈JUj=∅)
Examples
Consider a Standard Topology on real numbers(R,U) and a collection of subsets {Ui}i∈I⊂X and a collection of subsets {(0,n1]}n∈N.
Then, the collection have finite intersection property.
real numberR with usual order has the least upper bound property.
rational numbersR with usual order does not have the least upper bound property (∵(0,2)∩Q has no Supremum in Q).
Consider a Metric Space(X,T,d), and an open covering {Ui}i∈I⊂T.
If X is compact, then ∃δ>0s.t.∀B⊂Xs.t.diam(B)<δ,∃Ui∈{Ui}i∈Is.t.Ui⊃B
where δ is called a Lebesgue number for the covering {Ui}i∈I.
Consider a subspace topology (A,T′) of a Topological Space(X,T), and a point x∈A.
x∈A is an isolated point if the singleton set {x} is open in the Subspace Topology ({x}∈T′).
Or, equivalently, x∈A is an isolated point if ∃Nxs.t.Nx∩(A∖{x})=∅ where Nx is the Neighborhood of x.
Facts
Consider a nonempty CompactHausdorff SpaceX.
If X has no isolated point, then X is uncountable.
Consider a Product TopologyZ×Y of the two topological spaces(N,TX) and (Y,TY) where Y={a,b} and TY={∅,{a,b}}.
It is not Compact but limit point compact
∵ Each element {n}×{a} is a Limit Point of {n}×{b} and vice versa by the construction of topology TY
Consider a completeMetric Space(X,T,d) and a Contraction Map on it f:X→X.
Then there exists a unique point x∈X such that f(x)=x.
∃!x∈Xs.t.f(x)=x
where the point x is called the fixed point of f.
Let two Riemannian manifolds(M,gM) and (N,gN), and a Diffeomorphismf:M→N. Then f is called an isometry if the following condition holds:
∀p∈M,u,v∈Tp(M),gM(u,v)=gN(f(u),f(v))
Consider a Hausdorff Space(X,T).
X is locally compact if and only if given x∈X and NeighborhoodU of x, there exists another NeighborhoodV of x such that its ClosureVˉ is Compact and is included in U∀x∈X,∀U:=Nx,∃V:=Nxs.t.Vˉis compactandVˉ⊂U
Consider a locally compact Hausdorff Space(X,T) and a subset A⊂X.
A∈Tor(X∖A)∈T⟹Ais locally compact
Consider a Topological Space(X,T).
The space is countably compact if every countableopen covering (a collection of countableopen subsets {Ui}i∈N of X whose union is X) of X has a finite subcovering.
Consider a Topological Space(X,T).
The space is a Lindelof space if every open covering (a collection of open subsets {Ui}i∈I of X whose union is X) of X has a countable subcovering.
A Topological Space(X,T) is regular (T3) space if it satisfies T1 Axiom and for all disjoint x and Closed SetB in X, there exists disjoint open sets containing x and B respectively.
∀x∈X,(X∖B)∈Ts.t.x∩B=∅,∃U,V∈Ts.t.U∩V=∅,x∈U,B⊂V
A Topological Space(X,T) is completely regular (T321, T3.5 or Tychnoff) space if it satisfies T1 Axiom and for all disjoint x and Closed SetB in X, there exists a continuous real-valued function f:X→R which separates x and B.
∀x∈X,(X∖B)∈Ts.t.x∩B=∅,∃(f:X→R)∈C0s.t.f({x})={0},f(B)={1}
Facts
Consider a completely regular space (X,T).
Then the weak topology for X generated by the set of all boundedcontinuous functionsC(X,R)={(f:X→R)∈C0∣∀x∈X,∃M∈Rs.t.∣f(x)∣≤M} is given topology T.
A Topological Space(X,T) is normal (T4) space if it satisfies T1 Axiom and for all disjoint closed setsA,B in X, there exists disjoint open sets containing A and B respectively.
∀(X−A),(X−B)∈Ts.t.A∩B=∅,∃U,V∈Ts.t.U∩V=∅,A⊂U,B⊂V
Or equivalently, ∀A,Us.t.(X−A)∈T,A∈U∈T,∃V∈Ts.t.A⊂VandVˉ⊂U
Examples
A lower limit topology Tl generated by Bl={[a,b)⊂R∣a<b} on real numbers(R,Tl) is normal space.
A Topological Space(X,T) is completely normal (T5) space if it satisfies T1 Axiom and for all Separated SetsA,B⊂X, there exists disjoint open sets containing A and B respectively.
∀A,Bs.t.(A∩B=∅andA∩B=∅),∃U,V∈Ts.t.U∩V=∅,A⊂U,B⊂V
Consider a Normal Space(X,T) and disjoint closed subsets A,B⊂X of X.
Then, there exists a continuous map f:X→[0,1]s.t.f(x)={01x∈Ax∈B.
∀A,B⊂Xs.t.(X∖A),(X∖B)∈TandA∩B=∅⟹∃f∈C0(X,[0,1])s.t.f(x)={01x∈Ax∈B
Consider a Topological Space(X,T) and a subset A⊂X.
The subset A is a Gδ set if there exists a countable collection of open sets {Ui}i∈N whose intersection is A.
∃{Ui}i∈Ns.t.Ui∈Tand⋂i∈NUi=A
Given two differentiable manifoldsM and N, a differentiable map f:M→N is a diffeomorphism if it is a Bijective and its inverse f−1 is differentiable as well.
Consider a Hausdorff SpaceX and a CompactHausdorff SpaceY.
The Y is a compactification of X if X is contained in Y as a subspace such that Xˉ=Y (Dense Set).
Or equivalently (Y,e) is a compactification of X if there exists an embedding e:X↪Y whose image e(X) is a dense subspace of Y.
∃e:X↪Ys.t.e(X)=Y
Consider a set X, a metric space (M,d′), and the set of all boundedcontinuous functionsC(X,M) on X to M.
Then, the supremum metric d on X is defined as
d(f,g)=sup{d′(f(x),g(x))∣x∈X},∀f,g∈C(X,M)
Consider a Completely Regular Space(X,T). Let C(X,R) denote the set of all boundedcontinuous functions on X to R.
Then there exists a Compactificationβ(X) of X such that every element of C(X,R) can be uniquely extended to an element of C(β(X),R).
Where the compactification β(X) is called the Stone-Cech compactification of X.
Consider a Topological Space(X,T) and a collection of subsets A={Ai}i∈I of X.
The collection A is locally finite in X if every point x of X has a NeighborhoodUx∈T that intersects only finitely many elements of A.
∀x∈X,∃Nxs.t.∣Nx∩A∣<∞
where Nx is a Neighborhood of x.
Consider a Topological Space(X,T) and a collection of subsets B={Bi⊂X}i∈I of X.
The collection B is σ-locally finite if B in the union of countable collection of Locally Finite families {Bn}n∈N.
∃{Bn}n∈Ns.t.B=⋃n∈NBn
A Cauchy sequence is a Sequence whose elements become arbitrarily close to each other as the sequence progresses.
(an)s.t.∀ϵ>0,∃N∈Ns.t.(∀m,n∈Ns.t.m,n>N,∣am−an∣<ϵ)
For a Metric Space(X,d) there exists completeMetric Space(Y,d′) and an isometric embedding e:X→Y for which e(X) is a dense subset of Y. The space (Y,d′) is unique up to metric equivalence.
The space (Y,d′) is called the completion of (X,d).
Consider a Metric Space(X,T,d) and a positive Real Numberϵ>0.
A finite subset Aϵ⊂X is called an ϵ-net of X if for every point x∈X, there exists an element a∈Aϵ such that the distance between a and x is less than ϵ.
Aϵ⊂Xs.t.∣Aϵ∣<∞and∀x∈X,∃a∈Aϵs.t.d(x,a)<ϵ
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A metric topology is a Topology generated by a basis defined by a Metric . where is an -ball centered at .
Consider a Standard Topology on Real Number and a quotient map . The quotient topology on the set induced by is generated as
Topologist’s sine curve
Topologist’s comb
