Algebraic topology vs point-set topology
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IdeaTopology is one of the basic fields of mathematics. The term is also used for a particular structure in a topological space; see topological structure for that. The subject of topology deals with the expressions of continuity and boundary, and studying the geometric properties of (originally: metric) spaces and relations of subspaces, which do not change under continuous deformations, regardless to other (such as in their metric properties). Topology as a structure enables one to model continuity and convergence locally. More recently, in metric spaces, topologists and geometric group theorists started looking at asymptotic properties at large, which are in some sense dual to the standard topological structure and are usually referred to as coarse topology. There are many cousins of the concept of topological spaces, e.g. sites, locales, topoi, higher topoi, uniformity spaces and so on, which specialize or generalize some aspect or structure usually found in Top. One of the tools of topology, homotopy theory, has long since crossed the boundaries of topology and applies to many other areas, thanks to many examples and motivations as well as of abstract categorical frameworks for homotopy like Quillen model categories, Brown’s categories of fibrant objects and so on. IntroductionThe following gives a quick introduction to some of the core concepts and tools of topology: A detailed introduction is going to be at Introduction to Topology. ContinuityThe key idea of topology is to study spaces with “continuous maps” between them. The concept of continuity was made precise first in analysis, in terms of epsilontic analysis of open balls, recalled as def. below. Then it was realized that this has a more elegant formulation in terms of the more general concept of open sets, this is prop. below. Adopting the latter as the definition leads to the concept of topological spaces, def. below. First recall the basic concepts from analysis:
(metric space) A metric space is such that for all x,y,z∈Xx,y,z \in X:
Every normed vector space (V,|−|)(V, {\vert - \vert}) becomes a metric space according to def. by setting d(x,y)≔|x−y|. d(x,y) \coloneqq {\vert x-y \vert} \,.
(epsilontic definition of continuity)
 For (X,d X)(X,d_X) and (Y,d Y)(Y,d_Y) two metric spaces (def. ), then a function f:X⟶Y f \;\colon\; X \longrightarrow Y is said to be continuous at a point x∈Xx \in X if for every ϵ>0\epsilon \gt 0 there exists δ>0\delta\gt 0 such that d X(x,y)<δ⇒d Y(f(x),f(y))<ϵ d_X(x,y) \lt \delta \;\;\Rightarrow\;\; d_Y(f(x), f(y)) \lt \epsilon or equivalently such that f(B x ∘(δ))⊂B f(x) ∘(ϵ) f(\;B_x^\circ(\delta)\;) \;\subset\; B^\circ_{f(x)}(\epsilon) where B ∘B^\circ denotes the open ball (definition ). The function ff is called just continuous if it is continuous at every point x∈Xx \in X. We now reformulate this analytic concept in terms of the simple but important concept of open sets:
(open ball) Let (X,d)(X,d), be a metric space. Then for every element x∈Xx \in X and every ϵ∈ℝ +\epsilon \in \mathbb{R}_+ a positive real number, write B x ∘(ϵ)≔{y∈X|d(x,y)<ϵ} B^\circ_x(\epsilon) \;\coloneqq\; \left\{ y \in X \;\vert\; d(x,y) \lt \epsilon \right\} for the open ball of radius ϵ\epsilon around xx. The following picture shows a point xx, some open balls B iB_i containing it, and two of its neighbourhoods U iU_i:
(rephrasing continuity in terms of open sets) A function f:X→Yf \colon X \to Y between metric spaces (def. ) is continuous in the epsilontic sense of def. precisely if it has the property that its pre-images of open subsets of YY (in the sense of def. ) are open subsets of XX.
First assume that ff is continuous in the epsilontic sense. Then for O Y⊂YO_Y \subset Y any open subset and x∈f −1(O Y)x \in f^{-1}(O_Y) any point in the pre-image, we need to show that there exists a neighbourhood of xx in f −1(O Y)f^{-1}(O_Y). But by assumption there exists an open ball B x ∘(ϵ)B_x^\circ(\epsilon) with f(B x ∘(ϵ))⊂O Yf(B_x^\circ(\epsilon)) \subset O_Y. Since this is true for all xx, by definition this means that f −1(O Y)f^{-1}(O_Y) is open in XX. Conversely, assume that f −1f^{-1} takes open subsets to open subsets. Then for every x∈Xx \in X and B f(x) ∘(ϵ)B_{f(x)}^\circ(\epsilon) an open ball around its image, we need to produce an open ball B x ∘(δ)B_x^\circ(\delta) in its pre-image. But by assumption f −1(B f(x) ∘(ϵ))f^{-1}(B_{f(x)}^\circ(\epsilon)) contains a neighbourhood of xx which by definition means that it contains such an open ball around xx. Topological spacesTherefore we should pay attention to open subsets. It turns out that the following closure property is what characterizes the concept:
(closure properties of open sets in a metric space) The collection of open subsets of a metric space (X,d)(X,d) as in def. has the following properties: In particular
and
This motivates the following generalized definition: The following shows all the topologies on the 3-element set (up to permutation of elements)
It is now immediate to formally implement the
(continuous maps) A continuous function between topological spaces f:(X,τ X)→(Y,τ Y) f \colon (X, \tau_X) \to (Y, \tau_Y) is a function between the underlying sets, f:X⟶Y f \colon X \longrightarrow Y such that pre-images under ff of open subsets of YY are open subsets of XX. The simple definition of open subsets and the simple principle of continuity gives topology its fundamental and universal flavor. The combinatorial nature of these definitions makes topology closely related to formal logic (for more on this see at locale). Our motivating example now reads: One point of the general definition of “topological space” is that it admits constructions which intuitively should exist on “continuous spaces”, but which do not in general exist, for instance, as metric spaces: The above picture shows on the left the square (a topological subspace of the plane), then in the middle the resulting quotient topological space obtained by identifying two opposite sides (the cylinder), and on the right the further quotient obtained by identifying the remaining sides (the torus).
(product topological space)
For XX and YY two topological spaces, then the product topological space X×YX \times Y has and
These constructions of discrete topological spaces, quotient topological spaces, topological subspaces and of product topological spaces are simple examples of limits and of colimits of topological spaces. The category Top of topological spaces has the convenient property that all limits and colimits (over small diagrams) exist in it. (For more on this see at Top – Universal constructions.) HomeomorphismWith the objects (topological spaces) and the morphisms (continuous maps) of the category Top of topology thus defined, we obtain the concept of “sameness” in topology. To make this precise, one says that a morphism X→fY X \overset{f}{\to} Y in a category is an isomorphism if there exists a morphism going the other way around X⟵f −1Y X \overset{f^{-1}}{\longleftarrow} Y which is an inverse in the sense that f∘f −1=id Yandf −1∘f=id X. f \circ f^{-1} = id_Y \;\;\;\;\; and \;\;\;\;\; f^{-1} \circ f = id_X \,.
(open interval homeomorphic to the real line) The open interval (−1,1)(-1,1) is homeomorphic to all of the real line (−1,1)≃homeoℝ 1. (-1,1) \underset{homeo}{\simeq} \mathbb{R}^1 \,. An inverse pair of continuous functions is for instance given by f : ℝ 1 ⟶ (−1,+1) x ↦ x1+x 2 \array{ f &\colon& \mathbb{R}^1 &\longrightarrow& (-1,+1) \\ && x &\mapsto& \frac{x}{\sqrt{1+ x^2}} } and f −1 : (−1,+1) ⟶ ℝ 1 x ↦ x1−x 2. \array{ f^{-1} &\colon& (-1,+1) &\longrightarrow& \mathbb{R}^1 \\ && x &\mapsto& \frac{x}{\sqrt{1 - x^2}} } \,. Generally, every open ball in ℝ n\mathbb{R}^n (def. ) is homeomorphic to all of ℝ n\mathbb{R}^n.
(interval glued at endpoints is homeomorphic to the circle) As topological spaces, the interval with its two endpoints identified is homeomorphic (def. ) to the standard circle: [0,1] /(0∼1)≃homeoS 1. [0,1]_{/(0 \sim 1)} \;\; \underset{homeo}{\simeq} \;\; S^1 \,. More in detail: let S 1↪ℝ 2 S^1 \hookrightarrow \mathbb{R}^2 be the unit circle in the plane S 1={(x,y)∈ℝ 2,x 2+y 2=1} S^1 = \{(x,y) \in \mathbb{R}^2, x^2 + y^2 = 1\} equipped with the subspace topology (example ) of the plane ℝ 2\mathbb{R}^2, which itself equipped with its standard metric topology (example ). Moreover, let [0,1] /(0∼1) [0,1]_{/(0 \sim 1)} be the quotient topological space (example ) obtained from the interval [0,1]⊂ℝ 1[0,1] \subset \mathbb{R}^1 with its subspace topology by applying the equivalence relation which identifies the two endpoints (and nothing else). Consider then the function f:[0,1]⟶S 1 f \;\colon\; [0,1] \longrightarrow S^1 given by t↦(cos(2πt),sin(2πt)). t \mapsto (cos(2\pi t), sin(2\pi t)) \,. This has the property that f(0)=f(1)f(0) = f(1), so that it descends to the quotient topological space [0,1] ⟶ [0,1] /(0∼1) f↘ ↓ f˜ S 1. \array{ [0,1] &\overset{}{\longrightarrow}& [0,1]_{/(0 \sim 1)} \\ & {}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{\tilde f}} \\ && S^1 } \,. We claim that f˜\tilde f is a homeomorphism (definition ). First of all it is immediate that f˜\tilde f is a continuous function. This follows immediately from the fact that ff is a continuous function and by definition of the quotient topology (example ). So we need to check that f˜\tilde f has a continuous inverse function. Clearly the restriction of ff itself to the open interval (0,1)(0,1) has a continuous inverse. It fails to have a continuous inverse on [0,1)[0,1) and on (0,1](0,1] and fails to have an inverse at all on [0,1], due to the fact that f(0)=f(1)f(0) = f(1). But the relation quotiented out in [0,1] /(0∼1)[0,1]_{/(0 \sim 1)} is exactly such as to fix this failure. Similarly: The square [0,1] 2[0,1]^2 with two of its sides identified is the cylinder, and with also the other two sides identified is the torus: If the sides are identified with opposite orientation, the result is the Möbius strip:
\, Important examples of pairs of spaces that are not homeomorphic include the following: The proof of theorem is surprisingly hard, given how obvious the statement seems intuitively. It requires tools from a field called algebraic topology (notably Brouwer's fixed point theorem). We showcase some basic tools of algebraic topology now and demonstrate the nature of their usage by proving two very simple special cases of the topological invariance of dimension (prop. and prop. below). \, HomotopyWe have seen above that for n≥1n \geq 1 then the open ball B 0 ∘(1)B_0^\circ(1) in ℝ n\mathbb{R}^n is not homeomorphic to, notably, the point *=ℝ 0\ast = \mathbb{R}^0 (example , theorem ). Nevertheless, intuitively the nn-ball is a “continuous deformation” of the point, obtained as the radius of the nn-ball tends to zero. This intuition is made precise by observing that there is a continuous function out of the product topological space (example ) of the open ball with the closed interval η:[0,1]×B 0 ∘(1)⟶ℝ n \eta \colon [0,1] \times B_0^\circ(1) \longrightarrow \mathbb{R}^n which is given by rescaling: (t,x)↦t⋅x. (t,x) \mapsto t \cdot x \,. This continuously interpolates between the open ball and the point in that for t=1t = 1 then it restricts to the defining inclusion B 0 ∘(1)B_0^\circ(1), while for t=0t = 0 then it restricts to the map constant on the origin.
We may summarize this situation by saying that there is a diagram of continuous functions of the form B 0 ∘(1)×{0} ↓ ↘ x↦0 [0,1]×B 0 ∘(1) ⟶(t,x)↦t⋅x ℝ n ↑ ↗ inclusion B 0 ∘(1)×{1} \array{ B_0^\circ(1) \times \{0\} \\ \downarrow & \searrow^{\mathrlap{x \mapsto 0}} \\ [0,1] \times B_0^\circ(1) &\overset{(t,x) \mapsto t \cdot x}{\longrightarrow}& \mathbb{R}^n \\ \uparrow & \nearrow_{\mathrlap{inclusion}} \\ B_0^\circ(1) \times \{1\} } Such “continuous deformations” are called homotopies:
(homotopy) For f,g:X⟶Yf,g\colon X \longrightarrow Y two continuous functions between topological spaces X,YX,Y, then a (left) homotopy η:f⇒ Lg \eta \colon f \,\Rightarrow_L\, g is a continuous function η:X×I⟶Y \eta \;\colon\; X \times I \longrightarrow Y out of the product topological space (example ) of the open ball with the standard interval, such that this fits into a commuting diagram of the form
0×X (id,δ 0)↓ ↘ f [0,1]×X ⟶η Y (id,δ 1)↑ ↗ g {1}×X. \array{ {0} \times X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ [0,1] \times X &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ \{1\} \times X } \,.
(homotopy equivalence) A continuous function f:X⟶Yf \;\colon\; X \longrightarrow Y is called a homotopy equivalence if
η 1:f∘g⇒ Lid Y \eta_1 \;\colon\; f\circ g \Rightarrow_L id_Y and η 2:g∘f⇒ Lid X. \eta_2 \;\colon\; g\circ f \Rightarrow_L id_X \,. \, Connected componentsUsing the concept of homotopy one obtains the basic tool of algebraic topology, namely the construction of algebraic homotopy invariants of topological spaces. We introduce the simplest and indicate their use.
A homotopy between two points x,y:*→Xx,y \;\colon\; \ast \to X is a continuous path between these points. This construction is evidently compatible with composition, in that π 0(g∘f)=π 0(g)∘π 0(f) \pi_0(g \circ f) = \pi_0(g) \circ \pi_0(f) and it evidently is unital, in that π 0(id X)=id π 0(X). \pi_0(id_X) = id_{\pi_{0}(X)} \,. One summarizes this by saying that π 0\pi_0 is a functor from the category Top of topological spaces to the category Set of sets, denoted π 0:Top⟶Set. \pi_0 \;\colon\; Top \longrightarrow Set \,. An immediate but important consequence is this:
Since π 0\pi_0 is functorial, it immediately follows that it sends isomorphisms to isomorphisms, hence homeomorphisms to bijections: f∘g=idandg∘f=id ⇒ π 0(f∘g)=π 0(id)andπ 0(g∘f)=π 0(id) ⇔ π 0(f)∘π 0(g)=idandπ 0(g)∘π 0(f)=id. \begin{aligned} & f \circ g = id \;\;and\;\; g \circ f = id \\ \Rightarrow \;\;\;\;\;\;& \pi_0(f \circ g) = \pi_0(id) \;\;and \;\; \pi_0(g \circ f) = \pi_0(id) \\ \Leftrightarrow \;\;\;\;\;\; & \pi_0(f) \circ \pi_0(g) = id \;\;and \;\; \pi_0(g) \circ \pi_0(f) = id \end{aligned} \,. This means that we may use path connected components as a first “topological invariant” that allows us to distinguish some topological spaces. As an example for how this is being used, we have the following proof of a simple special case of the topological invariance of dimension (theorem ):
Assume there were a homeomorphism f:ℝ 1⟶ℝ 2 f \colon \mathbb{R}^1 \longrightarrow \mathbb{R}^2 we will derive a contradiction. If ff is a homeomorphism, then clearly so is its restriction to the topological subspaces (example ) obtained by removing 0∈ℝ 10 \in \mathbb{R}^1 and f(0)∈ℝ 2f(0) \in \mathbb{R}^2. f:(ℝ 1−{0})⟶(ℝ 2−{f(0)}). f \;\colon\; (\mathbb{R}^1-\{0\}) \longrightarrow (\mathbb{R}^2 - \{f(0)\}) \,. It follows that we would get a bijection of connected components between π 0(ℝ 1−{0})\pi_0(\mathbb{R}^1 - \{0\}) and π 0(ℝ 2−{f(0)})\pi_0(\mathbb{R}^2 - \{f(0)\}). But clearly the first set has two elements, while the second has just one: π 0(ℝ 1−{0})≠π 0(ℝ 2−{f(0)}). \pi_0(\mathbb{R}^1-\{0\}) \;\neq\; \pi_0(\mathbb{R}^2 - \{f(0)\}) \,. The key lesson of the proof of prop. is its strategy: Of course in practice one uses more sophisticated invariants than just π 0\pi_0. The next topological invariant after the connected components is the fundamental group: Fundamental group
(fundamental group) Let XX be a topological space and let x∈Xx \in X be a chosen point. Then write π 1(X,x)∈Grp \pi_1(X,x) \;\in\; Grp for, to start with, the set of homotopy classes of paths in XX that start and end at xx. Such paths are also called the continuous loops in XX based at xx.
This is called the fundamental group of XX at xx. The following picture indicates the four non-equivalent non-trivial generators of the fundamental group of the oriented surface of genus 2:
Again, this operation is functorial, now on the category Top */Top^{\ast/} of pointed topological spaces, whose objects are topological spaces equipped with a chosen point, and whose morphisms are continuous maps f:X→Yf \colon X \to Y that take the chosen basepoint of XX to that of YY: π 1:Top */⟶Grp. \pi_1 \;\colon\; Top^{\ast/} \longrightarrow Grp \,. As π 0\pi_0, so also π 1\pi_1 is a topological invariant. As before, we may use this to prove a simple case of the theorem of the topological invariance of dimension:
Assume there were such a homeomorphism ff; we will derive a contradiction. If ff is a homeomorphism, then so is its restriction to removing the origin from ℝ 2\mathbb{R}^2 and f(0)f(0) from ℝ 3\mathbb{R}^3: (ℝ 2−{0})⟶(ℝ 3−{f(0)}). (\mathbb{R}^2 - \{0\}) \longrightarrow (\mathbb{R}^3 - \{f(0)\}) \,. Thse two spaces are both path-connected, hence π 0\pi_0 does not distiguish them. But they do have different fundamental groups π 1\pi_1:
But since passing to fundamental groups is functorial, the same argument as in the proof of prop. shows that ff cannot be an isomorphism, hence not a homeomorphism. We now discuss a “dual incarnation” of fundamental groups, which often helps to compute them. Covering spaces
(covering space) A covering space of a topological space XX is a continuous map p:E→X p \colon E \to X such that there exists an open cover ⊔iU i→X\underset{i}{\sqcup}U_i \to X, such that restricted to each U iU_i then E→XE \to X is homeomorphic over U iU_i to the product topological space (example ) of U iU_i with the discrete topological space (example ) on a set F iF_i ⊔iU i×F i ⟶ E ↓ (pb) ↓ p ⊔iU i ⟶ X. \array{ \underset{i}{\sqcup} U_i \times F_i &\longrightarrow& E \\ \downarrow &(pb)& \downarrow^{\mathrlap{p}} \\ \underset{i}{\sqcup} U_i &\underset{}{\longrightarrow}& X } \,. For x∈U i⊂Xx \in U_i \subset X a point, then the elements in F x=F iF_x = F_i are called the leaves of the covering at xx.
(covering of circle by circle)
Regard the circle S 1={z∈ℂ||z|=1}S^1 = \{ z \in \mathbb{C} \;\vert\; {\vert z\vert} = 1 \} as the topological subspace of elements of unit absolute value in the complex plane. For k∈ℕk \in \mathbb{N}, consider the continuous function p≔(−) k:S 1⟶S 1 p \coloneqq (-)^k \;\colon\; S^1 \longrightarrow S^1 given by taking a complex number to its kkth power. This may be thought of as the result of “winding the circle kk times around itself”. Precisely, for k≥1k \geq 1 this is a covering space (def. ) with kk leaves at each point.
(covering of circle by real line)
Consider the continuous function exp(2πi(−)):ℝ 1⟶S 1 \exp(2 \pi i(-)) \;\colon\; \mathbb{R}^1 \longrightarrow S^1 from the real line to the circle, which,
We may think of this as the result of “winding the line around the circle ad infinitum”. Precisely, this is a covering space (def. ) with the leaves at each point forming the set ℤ\mathbb{Z} of natural numbers.
(action of fundamental group on fibers of covering) Let E⟶πXE \overset{\pi}{\longrightarrow} X be a covering space (def. ) Then for x∈Xx \in X any point, and any choice of element e∈F xe \in F_x of the leaf space over xx, there is, up to homotopy, a unique way to lift a representative path in XX of an element γ\gamma of the the fundamental group π 1(X,x)\pi_1(X,x) (def. ) to a continuous path in EE that starts at ee. This path necessarily ends at some (other) point ρ γ(e)∈F x\rho_\gamma(e) \in F_x in the same fiber. This construction provides a function ρ : F x×π 1(X,x) ⟶ F x (e,γ) ↦ ρ γ(e) \array{ \rho &\colon& F_x \times \pi_1(X,x) &\longrightarrow& F_x \\ && (e,\gamma) &\mapsto& \rho_\gamma(e) } from the Cartesian product of the leaf space with the fundamental group. This function is compatible with the group-structure on π 1(X,x)\pi_1(X,x), in that the following diagrams commute: F x×{const x} ⟶ F x×π 1(X,x) id↘ ↙ ρ F x(the neutral element, i.e. the constant loop, acts trivially) \array{ F_x \times \{const_x\} && \longrightarrow && F_x \times \pi_1(X,x) \\ & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{\rho}} \\ && F_x } \;\;\;\;\;\; \left( \array{ \text{the neutral element,} \\ \text{i.e. the constant loop,} \\ \text{acts trivially} } \right) and F x×π 1(X,x)×π 1(X,x) ⟶ρ×id F x×π 1(X,x) id×((−)⋅(−))↓ ↓ ρ F x×π 1(X,x) ⟶ρ F x(acting with two group elements is the same as first multiplying them and then acting with their product element). \array{ F_x \times \pi_1(X,x) \times \pi_1(X,x) &\overset{\rho \times id}{\longrightarrow}& F_x \times \pi_1(X,x) \\ {}^{\mathllap{id \times ((-)\cdot(-))}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ F_x \times \pi_1(X,x) &\underset{\rho}{\longrightarrow}& F_x } \;\;\;\;\;\; \left( \array{ \text{acting with two group elements } \\ \text{is the same as} \\ \text{first multiplying them} \\ \text{ and then acting with their product element} } \right) \,. One says that ρ\rho is an action or permutation representation of π 1(X,x)\pi_1(X,x) on F xF_x. For GG any group, then there is a category GSetG Set whose objects are sets equipped with an action of GG, and whose morphisms are functions which respect these actions. The above construction yields a functor Cov(X)⟶π 1(X,x)Set. Cov(X) \longrightarrow \pi_1(X,x) Set \,.
(three-sheeted covers of the circle)
There are, up to isomorphism, three different 3-sheeted covering spaces of the circle S 1S^1. The one from example for k=3k = 3. Another one. And the trivial one. Their corresponding permutation actions may be seen from the pictures on the right.
We are now ready to state the main theorem about the fundamental group. Except that it does require the following slightly technical condition on the base topological space. This condition is satisfied for all “reasonable” topological spaces:
(fundamental theorem of covering spaces) Let XX be a topological space which is path-connected (def. ), locally path connected (def. ) and semi-locally simply connected (def. ). Then for any x∈Xx \in X the functor Fib x:Cov(X)⟶π 1(X,x)Set. Fib_x \;\colon\; Cov(X) \overset{}{\longrightarrow} \pi_1(X,x) Set \,. from def. that describes the action of the fundamental group of XX on the set of leaves over xx has the following property: A functor with these two properties one calls an equivalence of categories: Cov(X)⟶≃π 1(X,x)Set. Cov(X) \overset{\simeq}{\longrightarrow} \pi_1(X,x) Set \,. This has some interesting implications: Every sufficiently nice topological space XX as above has a covering which is simply connected (def. ). This is the covering corresponding, under the fundamental theorem of covering spaces (theorem ) to the action of π 1(X)\pi_1(X) on itself. This is called the universal covering space X^→X\hat X \to X. The above theorem implies that the fundamental group itself may be recovered as the automorphisms of the universal covering space: π 1(X)≃Aut Cov /X(X^,X^). \pi_1(X) \simeq Aut_{Cov_{/X}}(\hat X, \hat X) \,.
(computing the fundamental group of the circle) The covering exp(2πi(−)):ℝ 1→S 1\exp(2\pi i(-)) \;\colon\; \mathbb{R}^1 \to S^1 from example is simply connected (def. ), hence must be the universal covering space, up to homeomorphism. It is fairly straightforward to see that the only homeomorphisms from ℝ 1\mathbb{R}^1 to itself over S 1S^1 are given by integer translations by n∈ℕ↪ℝn \in \mathbb{N} \hookrightarrow \mathbb{R}: ℝ 1 ⟶≃t↦t+n ℝ 1 exp(2πi(−))↘ ↙ exp(2πi(−)) S 1. \array{ \mathbb{R}^1 && \underoverset{\simeq}{t \mapsto t + n}{\longrightarrow} && \mathbb{R}^1 \\ & {}_{\mathllap{\exp(2 \pi i(-))}}\searrow && \swarrow_{\mathrlap{\exp(2 \pi i(-))}} \\ && S^1 } \,. Hence Aut Cov /S 1(S^ 1,S^ 1)≃ℤ Aut_{Cov_{/S^1}}(\hat S^1, \hat S^1) \simeq \mathbb{Z} and hence the fundamental group of the circle is the additive group of integers: π 1(S 1)≃ℤ. \pi_1(S^1) \simeq \mathbb{Z} \,. \, \, Basic factsCentral theoremsThe following is an (incomplete) list of available nnLab entries related to topology. Topological spaces
sphere
See also examples in topology. Manifolds and generalizationsAlgebraic topology and homotopy theory
Topological homotopy theory
Simplicial homotopy theory
Sheaves, stacks, cohomology
Non-commutative topology
Computational TopologyReferencesHistorical origins: Textbook accounts:
See also: With emphasis on category theoretic aspects of general topology:
See also
and see further references at algebraic topology. Lecture notes: Basic topology set up in intuitionistic mathematics is discussed in
See also:
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