For \(A\subseteq X\), the subspace topology is the coarsest topology on \(A\) for which the inclusion is continuous; the product topology on \(\prod_\alpha X_\alpha\) is the coarsest for which every projection \(\pi_\alpha\) is continuous; and for a surjection \(q\colon X\to Y\), the quotient topology is the finest on \(Y\) for which \(q\) is continuous. Hence \(f\colon Z\to\prod_\alpha X_\alpha\) is continuous if and only if every \(\pi_\alpha\circ f\) is continuous, and \(g\colon Y\to W\) is continuous if and only if \(g\circ q\) is continuous. For the box topology on \(\RR^\omega\) the first criterion fails: the diagonal map \(t\mapsto(t,t,\ldots)\) has continuous components and is not continuous, since the preimage of \(\prod_n(-1/n,1/n)\) is \(\{0\}\). For \(A\subseteq X\) and \(f\colon A\to Y\), the adjunction space \(Y\cup_fX\) is the quotient of \(X\sqcup Y\) by \(a\sim f(a)\) for \(a\in A\); attaching cells is the case \(X=D^n\), \(A=S^{n-1}\). For a group \(G\) acting on \(X\) by homeomorphisms, the orbit space \(X/G\) has the quotient topology from \(X\to X/G\).
Subspace topology (Tag D-LB2LS)
Let \((X, \tau)\) be a topological space and \(A\subseteq X\) a subset. The subspace topology on \(A\) is \[ \tau_A\coloneqq\ts{A\cap U \st U\in\tau}. \]
See [@Mun00].
Product and box topologies (Tag D-C5THN)
Let \((X_i)_{i\in I}\) be a family of topological spaces, let \(X \coloneqq \prod_{i\in I} X_i\), and for \(j\in I\) let \(p_j\colon X\to X_j\) be the projection. The box topology on \(X\) is the topology generated by the basis \[ \mathcal B_{\Box} \coloneqq \ts{ \prod_{i\in I} U_i \st U_i \text{ open in } X_i \text{ for every } i }. \] The product topology on \(X\) is the topology generated by the basis \[ \mathcal B_{\prod} \coloneqq \ts{ \prod_{i\in I} U_i \st U_i \text{ open in } X_i \text{ for every } i, \text{ and } U_i = X_i \text{ for all but finitely many } i }. \]
The product topology on \(X\) is the coarsest topology on \(X\) for which every projection \(p_j\) is continuous. It is the topology generated by the subbasis \(\ts{ \inverseof{p_j}(U) \st j\in I,\ U \text{ open in } X_j }\).
Give \(X\) the product topology. For every topological space \(Y\) and every family of continuous maps \(f_i\colon Y\to X_i\), there is a unique continuous map \(f\colon Y\to X\) with \(p_i\circ f = f_i\) for every \(i\in I\). Hence \((X, (p_i)_{i\in I})\) is the product of the \(X_i\) in \(\Top\):
Every element of \(\mathcal B_{\prod}\) lies in \(\mathcal B_{\Box}\), so the box topology is finer than the product topology. The two topologies coincide when \(I\) is finite.
For the box topology, the universal property of the product fails. Let \(X = \prod_{n\geq 1}\RR\) and let \(\Delta\colon\RR\to X\), \(\Delta(t) = (t,t,\ldots)\), whose components are all \(\id_\RR\). Then \(\Delta\) is continuous for the product topology, but not for the box topology: \(\prod_{n\geq1}(-1/n, 1/n)\) is open in the box topology and \(\inverseof{\Delta}\qty{\prod_{n\geq1}(-1/n,1/n)} = \ts{0}\) is not open in \(\RR\).
See [@Mun00].
Product topology on $X\times Y$ (Tag D-JKH35)
Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces. The product topology on \(X\times Y\) is the topology generated by the basis \[ \mathcal B\coloneqq\ts{U\times V \st U\in\tau_X,\ V\in\tau_Y}: \] a subset \(W\subseteq X\times Y\) is open if and only if it is a union of elements of \(\mathcal B\).
See [@Mun00].
Quotient map (Tag D-ITBUT)
Let \(X\) and \(Y\) be topological spaces. A map \(q\colon X\to Y\) is a quotient map if \(q\) is surjective and, for every \(U\subseteq Y\), \(U\) is open in \(Y\) if and only if \(\inverseof{q}(U)\) is open in \(X\).
The implication from \(U\) open to \(\inverseof{q}(U)\) open is continuity of \(q\); a quotient map is a continuous surjection that also satisfies the converse implication.
A surjective map \(q\colon X\to Y\) is a quotient map if and only if \(q\) is continuous and \(q(V)\) is open in \(Y\) for every open subset \(V\subseteq X\) that is saturated with respect to \(q\). The same holds with “open” replaced by “closed” throughout.
See [@Mun00].
Saturated subset (Tag D-KWWVL)
Let \(p\colon X\to Y\) be a surjective map. A subset \(U\subseteq X\) is saturated with respect to \(p\) if for every \(y\in Y\) with \(U\cap \inverseof{p}(y)\neq\emptyset\) we have \(\inverseof{p}(y)\subseteq U\).
A subset \(U\subseteq X\) is saturated with respect to \(p\) if and only if \(U = \inverseof{p}(B)\) for some \(B\subseteq Y\), in which case \(B = p(U)\).
See [@Mun00].
Attaching a space along a map (Tag D-HRU62)
Let \(X_0\) and \(X_1\) be topological spaces, \(A\subseteq X_1\) a subspace, and \(f\colon A\to X_0\) a continuous map. The space obtained by attaching \(X_1\) to \(X_0\) along \(f\) is the quotient space \[ X_0\sqcup_f X_1\coloneqq\qty{X_0\sqcup X_1}/\qty{a\sim f(a) \text{ for } a\in A}, \] where \(\sim\) is the equivalence relation generated by \(a\sim f(a)\).
Attaching an \(n\)-cell to \(X_0\) along \(\varphi\colon S^{n-1}\to X_0\) is the case \(X_1 = D^n\), \(A = S^{n-1}\), \(f = \varphi\). For a continuous map \(g\colon X\to Y\), the mapping cylinder of \(g\) is the case \(X_0 = Y\), \(X_1 = X\times[0,1]\), \(A = X\times\ts{1}\), \(f(x, 1) = g(x)\).
See [@Hat02].
Orbit space (Tag D-O5NN7)
Let \(G\) be a group acting on a topological space \(X\). For \(x,y\in X\) write \(x\sim y\) if there exists \(g\in G\) with \(g\cdot x=y\); this is an equivalence relation whose classes are the orbits of the action. Let \(X/G\coloneqq X/{\sim}\) be the set of orbits and \(q\colon X\to X/G\) the map sending \(x\) to its orbit. The orbit space is \(X/G\) with the quotient topology: \(U\subseteq X/G\) is open if and only if \(\inverseof{q}(U)\) is open in \(X\), so that \(q\) is a quotient map.