Subspaces and quotient spaces are the two basic ways to form new topological spaces from old ones: restrict the open sets, or identify points and push the topology forward.
Subspace topology
A subspace inherits exactly the open information visible from the ambient space.
Definition. If \(A\subseteq X\), the subspace topology on \(A\) is \(\mathcal T_A=\{A\cap U:U\text{ open in }X\}\).
Proposition. A map \(f:Y\to A\) is continuous into the subspace \(A\) if and only if the composite \(Y\to A\hookrightarrow X\) is continuous.
Proof. If \(f:Y\to A\) is continuous, composing with the inclusion is continuous. Conversely, if the composite \(i\circ f:Y\to X\) is continuous and \(A\cap U\) is open in \(A\), then \(f^{-1}(A\cap U)=(i\circ f)^{-1}(U)\), open in \(Y\).
Proposition. For \(B\subseteq A\subseteq X\), the closure of \(B\) in \(A\) is \(A\cap\overline B^X\).
Proof. The set \(A\cap\overline B^X\) is closed in \(A\) and contains \(B\). If \(C\) is closed in \(A\) and contains \(B\), then \(C=A\cap F\) for some closed \(F\subseteq X\) with \(B\subseteq F\), so \(\overline B^X\subseteq F\) and \(A\cap\overline B^X\subseteq C\).
Quotient topology
A quotient topology is the finest topology on the set of identified points that makes the projection continuous.
Definition. Let \(q:X\to Y\) be a surjective map. The quotient topology on \(Y\) is defined by declaring \(V\subseteq Y\) open if and only if \(q^{-1}(V)\) is open in \(X\). A map with this topology is a quotient map.
Proposition. The quotient topology is a topology, and \(q:X\to Y\) is continuous by construction.
Proof. The empty set and \(Y\) pull back to \(\varnothing\) and \(X\). Inverse images preserve arbitrary unions and finite intersections, so the declared open sets are closed under the topology operations. Since an open set \(V\subseteq Y\) is defined by \(q^{-1}(V)\) being open, \(q\) is continuous.
Universal property of quotients
The quotient topology is best understood by how maps out of the quotient are tested.
Theorem. Let \(q:X\to Y\) be a quotient map and \(Z\) a topological space. A map \(g:Y\to Z\) is continuous if and only if \(g\circ q:X\to Z\) is continuous.
Proof. If \(g\) is continuous, then \(g\circ q\) is continuous. Conversely, suppose \(g\circ q\) is continuous. For an open set \(W\subseteq Z\), \((g\circ q)^{-1}(W)=q^{-1}(g^{-1}(W))\) is open in \(X\). By the definition of quotient topology, \(g^{-1}(W)\) is open in \(Y\), so \(g\) is continuous.
Corollary. To define a continuous map on a quotient \(X/{\sim}\), it is enough to define a continuous map on \(X\) that is constant on equivalence classes.
Proof. Such a map factors uniquely through the quotient set. The universal property then gives continuity of the induced map.
Saturated sets and common pitfalls
A quotient can destroy separation properties if equivalence classes are not well behaved.
Definition. A subset \(A\subseteq X\) is saturated with respect to \(q:X\to Y\) if \(A=q^{-1}(q(A))\), equivalently if it is a union of fibers of \(q\).
Proposition. For a quotient map \(q:X\to Y\), open subsets of \(Y\) correspond exactly to open saturated subsets of \(X\).
Proof. If \(V\subseteq Y\) is open, then \(q^{-1}(V)\) is open and saturated. Conversely, if \(A\subseteq X\) is open and saturated, then \(A=q^{-1}(q(A))\); by the quotient definition, \(q(A)\) is open in \(Y\).
Example. Identifying the endpoints of \([0,1]\) gives a circle. Identifying a nonclosed dense subset to a point may produce a quotient that is not Hausdorff.
Quotients can destroy separation
Counterexample. Let \(\mathbb Q\subset\mathbb R\) be collapsed to one point. In the quotient \(\mathbb R/\mathbb Q\), every neighborhood of the collapsed point has dense preimage and meets every saturated neighborhood of every irrational class. The quotient is not Hausdorff. Thus a quotient of a Hausdorff space need not be Hausdorff.
Proposition. If \(X\) is compact Hausdorff and an equivalence relation \[ R=\{(x,y):x\sim y\}\subset X\times X \] is closed, then \(X/{\sim}\) is compact Hausdorff.
Closedness of \(R\) separates inequivalent classes by saturated neighborhoods; compactness makes the saturation argument finite. This criterion explains why gluing closed faces of a compact polygon produces well-behaved surfaces, while arbitrary identifications can produce non-Hausdorff spaces.
Universal property of a quotient
If \(q:X\to X/{\sim}\), a map \(\overline f:X/{\sim}\to Y\) is continuous exactly when \(\overline f\circ q\) is continuous. Therefore a continuous \(f:X\to Y\) descends uniquely precisely when it is constant on equivalence classes. This is the reliable method for defining maps out of glued spaces.