A space \(X\) is connected if its only subsets that are both open and closed are \(\varnothing\) and \(X\), and path-connected if any two points are joined by a continuous path \([0,1]\to X\). Continuous images of connected spaces are connected, continuous images of path-connected spaces are path-connected, and path-connected spaces are connected; the topologist’s sine curve is connected and not path-connected. A homotopy from \(f\colon X\to Y\) to \(g\colon X\to Y\) is a continuous \(H\colon X\times[0,1]\to Y\) with \(H(x,0)=f(x)\) and \(H(x,1)=g(x)\), and homotopy is an equivalence relation on continuous maps \(X\to Y\). The spaces \(X\) and \(Y\) are homotopy equivalent if there are continuous \(f\colon X\to Y\) and \(g\colon Y\to X\) with \(g\circ f\simeq\id_X\) and \(f\circ g\simeq\id_Y\); homeomorphic spaces are homotopy equivalent, and \(\RR^n\) is homotopy equivalent to a point. A subspace \(A\subseteq X\) is a deformation retract if there is a homotopy from \(\id_X\) to a retraction \(X\to A\) that fixes every point of \(A\) at all times. A space is contractible if \(\id_X\) is homotopic to a constant map; any two maps from a space into a contractible space are homotopic.
Characterizations of connectedness (Tag D-YO6NZ)
Let \(X\) be a topological space. A separation of \(X\) is a pair \((U, V)\) of disjoint nonempty open subsets of \(X\) with \(X = U\union V\). The space \(X\) is disconnected if it has a separation, and connected otherwise.
For a topological space \(X\), the following are equivalent:
\(X\) is connected.
The only subsets of \(X\) that are both open and closed are \(\emptyset\) and \(X\).
Every continuous map from \(X\) to the discrete space \(\ts{0, 1}\) is constant.
Let \(X\) be a topological space and \(Y\subseteq X\) a subspace. Then \(Y\) is disconnected if and only if there exist disjoint nonempty subsets \(A, B\subseteq Y\) with \(Y = A\union B\) such that neither contains a limit point of the other, that is, \(\cl_X(A)\intersect B = \emptyset\) and \(A\intersect\cl_X(B) = \emptyset\).
The subspace \(\QQ\subseteq\RR\) is disconnected, and every connected subspace of \(\QQ\) has at most one point, so the connected components of \(\QQ\) are its singletons. For rationals \(p < q\) and an irrational \(r\) with \(p<r<q\), the sets \(\QQ\intersect(-\infty, r)\) and \(\QQ\intersect(r, \infty)\) form a separation of \(\QQ\).
[@Mun00].
Connected components (Tag D-FAEYE)
Let \(X\) be a topological space. For \(x, y\in X\) write \(x\sim y\) if there is a connected subspace \(C\subseteq X\) with \(x, y\in C\). The equivalence classes of \(\sim\) are the connected components of \(X\).
The relation \(\sim\) is an equivalence relation on \(X\).
See [@Mun00].
Path-connected space (Tag D-X73EB)
Let \(X\) be a topological space and \(I = [0,1]\). The space \(X\) is path connected if for every pair of points \(x\neq y\) in \(X\) there exists a path \(f\colon I \to X\) with \(f(0) = x\) and \(f(1) = y\).
Path components (Tag D-VZFJQ)
Let \(X\) be a topological space. For \(x, y\in X\), put \(x\sim y\) if there exists a path-connected subspace \(U\subseteq X\) with \(x, y\in U\). The equivalence classes of \(\sim\) are the path components of \(X\).
The relation \(\sim\) is an equivalence relation on \(X\), and \(x\sim y\) if and only if there is a path in \(X\) from \(x\) to \(y\).
[@Mun00].
Locally connected space (Tag D-GYBZ2)
A topological space \(X\) is locally connected if for every \(x\in X\) and every neighborhood \(U\) of \(x\) there is a connected open set \(V\) with \(x\in V\subseteq U\).
See [@Mun00].
Locally path-connected space (Tag D-7JSLO)
A topological space \(X\) is locally path-connected if for every \(x\in X\) and every neighborhood \(U\) of \(x\) there exists a path-connected open set \(V\) with \(x\in V\subseteq U\).
Equivalently, \(X\) has a basis consisting of path-connected open sets.
[@Mun00].
Homotopic maps (Tag D-Z7I7F)
Let \(X\) and \(Y\) be topological spaces, \(I = [0,1]\), and \(f, g\colon X\to Y\) continuous maps. The maps \(f\) and \(g\) are homotopic, written \(f\homotopic g\), if there is a continuous map \(H\colon X\cross I \to Y\) with \(H(x, 0) = f(x)\) and \(H(x, 1) = g(x)\) for all \(x\in X\); such a map \(H\) is a homotopy from \(f\) to \(g\). Two paths \(\gamma, \eta\colon I\to X\) with \(\gamma(0) = \eta(0)\) and \(\gamma(1) = \eta(1)\) are homotopic rel endpoints if there is a continuous map \(H\colon I\cross I\to X\) with \(H(s, 0) = \gamma(s)\), \(H(s, 1) = \eta(s)\), \(H(0,t) = \gamma(0)\), and \(H(1,t) = \gamma(1)\) for all \(s, t\in I\).
Homotopy is an equivalence relation on the set of continuous maps \(X\to Y\), and homotopy rel endpoints is an equivalence relation on the set of paths in \(X\) with given endpoints. The set of homotopy classes of continuous maps \(X\to Y\) is denoted \([X,Y]\).
The fundamental group \(\pi_1(X, x_0)\) is the set of classes of loops at \(x_0\) under homotopy rel endpoints.
[@Hat02]; [@Mun00].
Homotopy class (Tag D-IZI3T)
Let \(X\) and \(Y\) be topological spaces. The homotopy class \([f]\) of a continuous map \(f\colon X\to Y\) is its equivalence class under the relation \(\simeq\) of being homotopic. The set of homotopy classes of continuous maps \(X\to Y\) is \[ [X, Y]\coloneqq\Hom_{\Top}(X, Y)/\simeq. \]
If \(f, g\colon X\to Y\) are homotopic, then \(f_* = g_*\colon H_n(X)\to H_n(Y)\) for every \(n\).
For a based space \((Y, y_0)\) and \(n\geq 1\), the homotopy group \(\pi_n(Y, y_0)\) is the set \([(S^n, s_0), (Y, y_0)]\) of homotopy classes of based maps under homotopies fixing the basepoint.
See [@Hat02].
Nullhomotopic map (Tag D-MGRZP)
Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is nullhomotopic if it is homotopic to a constant map: there are \(y_0\in Y\) and a continuous map \(F\colon X\times[0,1]\to Y\) with \(F(x, 0) = f(x)\) and \(F(x, 1) = y_0\) for all \(x\in X\).
If \(f\colon X\to Y\) is nullhomotopic, then \(f_*\colon\tilde H_n(X)\to\tilde H_n(Y)\) is zero for every \(n\).
See [@Hat02].
Homotopy equivalence (Tag D-HFR32)
Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is a homotopy equivalence if there is a continuous map \(g\colon Y\to X\) with \(f\circ g\simeq\id_Y\) and \(g\circ f\simeq\id_X\), where \(\simeq\) denotes homotopy. Such a map \(g\) is a homotopy inverse of \(f\). The spaces \(X\) and \(Y\) are homotopy equivalent, or have the same homotopy type, written \(X\simeq Y\), if there is a homotopy equivalence \(X\to Y\).
Every homeomorphism is a homotopy equivalence, and the converse fails: for \(n\geq 1\) the inclusion \(\ts{0}\hookrightarrow\RR^n\) is a homotopy equivalence, with homotopy inverse the constant map \(\RR^n\to\ts{0}\) and homotopy \(H(x, t) = tx\) from the constant map to \(\id_{\RR^n}\), but \(\ts{0}\) and \(\RR^n\) are not homeomorphic.
Contractible space (Tag D-K43GA)
A topological space \(X\) is contractible if \(\id_X\) is nullhomotopic: there is a point \(x_0\in X\) such that \(\id_X\) is homotopic to the constant map \(c\colon X\to X\), \(c(x) = x_0\).
A topological space \(X\) is contractible if and only if \(X\) is homotopy equivalent to a one-point space \(\ts{x_0}\), that is, there are continuous maps \(f\colon X\to\ts{x_0}\) and \(g\colon\ts{x_0}\to X\) with \(f\circ g = \id_{\ts{x_0}}\) and \(g\circ f\simeq\id_X\).
See [@Hat02].
Retract (Tag D-NCLVD)
Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(\iota\colon A\injects X\) the inclusion. A retraction of \(X\) onto \(A\) is a continuous map \(r\colon X\to A\) with \(r\circ\iota=\id_A\), that is, \(r(a)=a\) for every \(a\in A\):
The subspace \(A\) is a retract of \(X\) if a retraction of \(X\) onto \(A\) exists.
Let \(A\subseteq X\) be a subspace with inclusion \(\iota\colon A\injects X\).
Every retraction \(r\colon X\to A\) is surjective.
Continuous maps \(r\colon X\to A\) with \(r\circ\iota=\id_A\) correspond bijectively to continuous maps \(\rho\colon X\to X\) with \(\rho(X)=A\) and \(\rho(a)=a\) for all \(a\in A\), via \(\rho=\iota\circ r\).
If \(r\) is a retraction and \(a_0\in A\), then \(r_*\circ\iota_* = \id\) on \(\pi_1(A, a_0)\) and on \(H_n(A)\), so \(\iota_*\colon \pi_1(A, a_0)\to\pi_1(X, a_0)\) and \(\iota_*\colon H_n(A)\to H_n(X)\) are injective. For every point \(x_0\) of a topological space \(X\), the subspace \(\ts{x_0}\) is a retract of \(X\), via the constant map \(X\to\ts{x_0}\).
Deformation retraction (Tag D-6UHU7)
Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(\iota\colon A\injects X\) the inclusion. A deformation retraction of \(X\) onto \(A\) is a homotopy \(F\colon X\cross I\to X\), written \(F_t \coloneqq F(\wait, t)\), such that for all \(t\in I\): \[ F_0 = \id_X, \qquad F_1(X) \subseteq A, \qquad F_t(a) = a \text{ for all } a\in A .\] The subspace \(A\) is a deformation retract of \(X\) if there exists a deformation retraction of \(X\) onto \(A\).
If \(F\) is a deformation retraction of \(X\) onto \(A\), then \(r\coloneqq F_1\colon X\to A\) is a retraction, \(r\circ\iota = \id_A\), and \(F\) is a homotopy from \(\id_X\) to \(\iota\circ r\). Hence \(\iota\) is a homotopy equivalence with homotopy inverse \(r\).
Two spaces \(X\) and \(Y\) are homotopy equivalent if and only if there exists a space \(Z\) containing both \(X\) and \(Y\) as deformation retracts; for a homotopy equivalence \(f\colon X\to Y\), the mapping cylinder of \(f\) is such a \(Z\).
[@Hat02].
Deformation and deformation retraction (Tag D-UH3L5)
Let \(X\) be a topological space and \(I=[0,1]\). A deformation of \(X\) is a family of maps \(f_t\colon X\to X\), \(t\in I\), such that \(f_0=\id_X\) and \((x,t)\mapsto f_t(x)\) is a continuous map \(X\times I\to X\). For a subspace \(A\subseteq X\), a deformation retraction of \(X\) onto \(A\) is a deformation \((f_t)_{t\in I}\) of \(X\) with \(f_1(X)=A\) and \(f_t(a)=a\) for all \(a\in A\) and \(t\in I\) [@Hat02].
If \(X\) deformation retracts onto \(A\), then the inclusion \(A\injects X\) is a homotopy equivalence [@Hat02].
Maps into contractible spaces are nullhomotopic (Tag T-TQ4J3)
Let \(X\) be a contractible space and \(Y\) any space. Then every continuous map \(f\colon Y \to X\) is nullhomotopic.