A topology on a set \(X\) is a family of subsets of \(X\), the open sets, that contains \(\varnothing\) and \(X\) and is closed under arbitrary unions and finite intersections. A basis \(\mathcal B\) generates the topology whose open sets are the unions of elements of \(\mathcal B\). Closed sets are complements of open sets; the closure \(\overline A\) of \(A\) is the smallest closed set containing \(A\), and \(x\in\overline A\) if and only if every neighbourhood of \(x\) meets \(A\). A map \(f\colon X\to Y\) is continuous if the preimage of every open set is open, equivalently if the preimage of every basis element is open, equivalently if the preimage of every closed set is closed, equivalently if \(f(\overline A)\subseteq\overline{f(A)}\) for every \(A\subseteq X\). A map is open if it sends open sets to open sets, and closed if it sends closed sets to closed sets. A homeomorphism is a continuous bijection with continuous inverse; a continuous bijection is a homeomorphism if and only if it is open.
Topology via open sets and closed sets (Tag D-2TZAI)
Let \(X\) be a set. A topology on \(X\) is a collection \(\tau\) of subsets of \(X\) such that
\(\emptyset \in \tau\) and \(X\in \tau\);
\(\Union_{i\in I} U_i \in \tau\) for every family \((U_i)_{i\in I}\) of elements of \(\tau\);
\(U_1\intersect \cdots \intersect U_n \in \tau\) for all \(n\geq 1\) and \(U_1, \ldots, U_n\in\tau\).
The pair \((X, \tau)\) is a topological space, the elements of \(\tau\) are its open sets, and a subset \(F\subseteq X\) is closed if \(X\sm F\) is open.
By De Morgan’s laws, a collection \(\mathcal F\) of subsets of \(X\) is the collection of closed sets of a topology on \(X\) if and only if \(\emptyset, X\in\mathcal F\), \(\mathcal F\) is closed under arbitrary intersections, and \(\mathcal F\) is closed under finite unions; the topology is then \(\ts{X\sm F \st F\in\mathcal F}\).
Basis for a topology (Tag D-WKURJ)
Let \(X\) be a set. A collection \(\mcb\) of subsets of \(X\) is a basis for a topology on \(X\) if
every \(x\in X\) lies in some \(B\in\mcb\), and
whenever \(x \in B_1 \intersect B_2\) with \(B_1, B_2\in\mcb\), there exists \(B_3\in\mcb\) with \(x\in B_3 \subseteq B_1 \intersect B_2\).
The topology generated by \(\mcb\) is the collection of all \(U\subseteq X\) such that for each \(x\in U\) there exists \(B\in\mcb\) with \(x\in B \subseteq U\).
Let \(\mcb\) be a basis for a topology on a set \(X\). The topology generated by \(\mcb\) is a topology on \(X\), and its open sets are exactly the unions of elements of \(\mcb\).
[@Mun00].
Closed set (Tag D-UI7ZL)
Let \(X\) be a topological space. A subset \(U \subseteq X\) is closed in \(X\) if its complement \(X\sm U\) is open in \(X\). If \(Y \subseteq X\) carries the subspace topology and \(U \subseteq Y\), then \(U\) is closed in the subspace \(Y\) if \(Y\sm U\) is open in \(Y\).
Let \(X\) be a topological space and \(U\subseteq X\).
\(U\) is closed in \(X\) if and only if every limit point of \(U\) in \(X\) lies in \(U\).
\(U\) is closed in \(X\) if and only if \(\cl_X(U) = U\), where \(\cl_X(U)\) is the closure of \(U\) in \(X\).
If \(Y\subseteq X\) is a subspace and \(U\subseteq Y\), then \(U\) is closed in \(Y\) if and only if \(U = Y \intersect V\) for some closed subset \(V\) of \(X\).
If \(Y\subseteq X\) is a subspace and \(U\subseteq Y\), then the closure of \(U\) in \(Y\) is \(\cl_Y(U) = \cl_X(U)\intersect Y\).
[@Mun00].
Closure of a set (Tag D-ASXW6)
Let \(X\) be a topological space and \(U\subseteq X\). The closure of \(U\) in \(X\) is the intersection of all closed subsets of \(X\) containing \(U\), \[ \cl_X(U) \coloneqq \Intersect_{\substack{B\supseteq U \\ B \text{ closed in } X}} B ,\] also written \(\bar U\).
If \(Y\subseteq X\) is a subspace containing \(U\), the closure of \(U\) in \(Y\) is \(\cl_Y(U) = \cl_X(U) \intersect Y\) [@Mun00]. A point \(x\in X\) lies in \(\cl_X(U)\) if and only if every neighborhood of \(x\) in \(X\) intersects \(U\) [@Mun00].
Limit point (Tag D-Y6JAS)
Let \(X\) be a topological space, \(A\subseteq X\), and \(x\in X\). The point \(x\) is a limit point of \(A\) if every neighborhood of \(x\) meets \(A\sm\ts{x}\). A limit point is also called an accumulation point.
A point \(x\in X\) is a limit point of \(A\subseteq X\) if and only if \(x\in \cl_{X}(A\sm\ts{x})\), the closure of \(A\sm\ts{x}\) in \(X\).
Interior point and interior (Tag D-5VCMJ)
Let \(X\) be a topological space and \(A\subseteq X\). A point \(p\in A\) is an interior point of \(A\) if there exists a neighborhood \(U\) of \(p\) with \(U\subseteq A\). The interior \(A^\circ\) of \(A\) is the set of interior points of \(A\).
Boundary of a subset (Tag D-3KS2F)
Let \(X\) be a topological space and \(A\subseteq X\). The boundary of \(A\) is \(\bd A \coloneqq \cl_X(A) \sm A^\circ\), where \(\cl_X(A)\) is the closure of \(A\) and \(A^\circ\), its interior, is the set of interior points of \(A\).
A point \(p\in X\) lies in \(\bd A\) if and only if every neighborhood of \(p\) intersects both \(A\) and \(X\sm A\).
Neighborhood (Tag D-JMRPA)
Let \(X\) be a topological space and \(x\in X\). A neighborhood of \(x\) is an open set \(U\subseteq X\) with \(x\in U\).
See [@Mun00].
Continuous map (Tag D-AEAAD)
Let \(X\) and \(Y\) be topological spaces. A map \(f\colon X\to Y\) is continuous if \(\inverseof{f}(U)\) is open in \(X\) for every open set \(U \subseteq Y\).
Characterizations of continuous maps (Tag T-FA6VI)
Let \(f\colon X\to Y\) be a map of topological spaces. The following are equivalent:
\(f\) is continuous;
\(f(\cl_X(A)) \subseteq \cl_Y(f(A))\) for every \(A\subseteq X\);
\(\inverseof{f}(B)\) is closed in \(X\) for every closed \(B\subseteq Y\);
for each \(x\in X\) and each neighborhood \(V\) of \(f(x)\), there is a neighborhood \(U\) of \(x\) such that \(f(U) \subseteq V\)
[@Mun00].
Open and closed maps (Tag D-CTGON)
Let \(X\) and \(Y\) be topological spaces and \(f\colon X\to Y\) a function. Then \(f\) is an open map if \(f(U)\) is open in \(Y\) for every open \(U\subseteq X\), and \(f\) is a closed map if \(f(C)\) is closed in \(Y\) for every closed \(C\subseteq X\).
Homeomorphism (Tag D-9KQZT)
Let \(X\) and \(Y\) be topological spaces. A bijection \(f\colon X\to Y\) is a homeomorphism if both \(f\) and \(\inverseof{f}\colon Y\to X\) are continuous. The spaces \(X\) and \(Y\) are homeomorphic, written \(X\cong Y\), if there exists a homeomorphism \(X\to Y\).
For a bijection \(f\colon X\to Y\) the following are equivalent: \(f\) is a homeomorphism; \(f\) is continuous and open; for every \(U\subseteq X\), \(U\) is open in \(X\) if and only if \(f(U)\) is open in \(Y\). A homeomorphism therefore induces a bijection between the topologies of \(X\) and \(Y\), and every property defined in terms of open sets holds for \(X\) if and only if it holds for \(Y\). An injective continuous map \(f\colon X\to Y\) that is a homeomorphism onto \(f(X)\) with the subspace topology is a topological embedding.
[@Mun00].
Topological embedding (Tag D-BCNUH)
Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is a topological embedding if it is injective and the map \(X\to f(X)\), \(x\mapsto f(x)\), is a homeomorphism, where \(f(X)\) carries the subspace topology.
Comparability of topologies (Tag D-KW52R)
Let \(\tau_1\) and \(\tau_2\) be topologies on a set \(X\). The topology \(\tau_1\) is finer than \(\tau_2\) if \(\tau_2\subseteq\tau_1\), and coarser than \(\tau_2\) if \(\tau_1\subseteq\tau_2\). The topologies \(\tau_1\) and \(\tau_2\) are comparable if \(\tau_1\subseteq\tau_2\) or \(\tau_2\subseteq\tau_1\).
See [@Mun00].