What the theory of these constructions still owes at this point in the reading order: preservation of quotient maps under products, separation beyond Hausdorff, and the invariants that tell the glued spaces apart

This page builds three constructions, the product, the coproduct and the quotient, proves the characteristic property of each, and develops further the subspace topology introduced earlier in the reading order. Four questions that a reader will ask immediately are deliberately left open, and this remark records which they are and why. Each is a question whose answer needs vocabulary that is not available at this point in the reading order; none of them is a defect of the constructions.

1. Is a product of quotient maps a quotient map? If $q : X \to Y$ and $q' : X' \to Y'$ are quotient maps (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection), is the map $q \times q' : X \times X' \to Y \times Y'$ (The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space) a quotient map? It is a continuous surjection, and it is a quotient map when both $q$ and $q'$ are open: the image of a basic open box $U \times U'$ under $q \times q'$ is $q[U] \times q'[U']$ (The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space), so $q \times q'$ is then an open continuous surjection, and an open continuous surjection is a quotient map (A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps, clause 1). That both factors are open is not a convenience of the argument: one of the two being open is not enough, and the standard counterexample is of exactly that shape, its second factor being an identity map, which is open. In general, then, the answer is no, and neither the standard counterexample nor the standard positive theorem is stated here. The positive theorem takes the form "if $q$ is a quotient map and $Z$ is suitably small, then $q \times \mathrm{id}_Z$ is a quotient map", and the smallness condition it needs is a compactness condition, which is later in the reading order. The counterexample that shows some such condition is necessary is a nested construction over an enumeration of $\mathbb{Q}$, out of proportion to what this page uses; nothing on this page or its companion depends on either.

2. Separation beyond the Hausdorff condition. Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not is introduced here in its minimal form and for one purpose only: to state and refute the claim that a quotient of a Hausdorff space is Hausdorff. The Hausdorff condition is the second of a graded family of separation conditions, and the questions that family raises, in particular which of its members are hereditary in the sense of Hereditary, open-hereditary and closed-hereditary properties of topological spaces and which are preserved by products, are not available at this point in the reading order. No statement on this page anticipates any of them, and in particular nothing here asserts or denies that any separation condition other than the Hausdorff one is hereditary.

3. The invariants that would distinguish the glued spaces. The adjunction space $Y \cup_f X$ glued along a continuous map, and, for a nonempty space, the cone and the suspension as quotients of $X \times [0,1]$ builds the adjunction space, the cone and the suspension, and the companion page builds $\mathbb{R}/\mathbb{Z}$, the torus, the cylinder and the Mobius band as quotients of an interval or a square. These are constructions, not classifications. Deciding that two of them are not homeomorphic requires an invariant, and the standard invariants for these particular spaces are connectedness, compactness and the homotopy notions. None of the three is available here in the form the question needs: connectedness and compactness are developed earlier in the reading order only for subsets of $\mathbb{R}$ and for metric spaces, neither of which covers a quotient that has not been shown metrizable, and the homotopy notions are developed only later in the reading order. Accordingly no item on this page or its companion claims that two of these spaces differ; where two constructions are shown to agree, an explicit homeomorphism is exhibited.

4. Metrizability of a product. Every subspace of a metrizable space is metrizable and every subspace of a first countable space is first countable, the metric case being the subspace metric already identified with the subspace topology shows that metrizability passes to subspaces. Whether it passes to products is a different question, and this page answers it only in named instances, never in general: For $n \ge 1$ the product topology on $n$ copies of the usual topology of $\mathbb{R}$ is the metric topology of $d_\infty$ on $\mathbb{R}^n$, and hence also of $d_1$ and $d_2$, so $\mathbb{R}^n$ as a product and $\mathbb{R}^n$ as a metric space are one space metrises the product of $n$ copies of $\mathbb{R}$, and the companion page metrises the Hilbert cube $[0,1]^{\mathbb{N}}$ by the explicit $\sum_k |x_k - y_k| / 2^{\,k+1}$, while its identification of $\{0,1\}^{\mathbb{N}}$ with the Cantor set metrises that product too, as a by-product of a homeomorphism rather than as an aim. No general theorem about products of metrizable spaces is stated here, in any number of factors, and the reader should not read any of these instances as more than the single instance it is.

One thing that is settled, and is worth separating from the four above. The coproduct raises no such question: a disjoint union of spaces is described completely by its traces (The disjoint union (coproduct) $\bigsqcup_i X_i$ with the final topology of the canonical injections: a set is open exactly when each of its traces is), maps out of it are described completely by their restrictions (A map out of a disjoint union is continuous iff each of its restrictions is; the canonical injections are open and closed embeddings; and each summand is clopen in the union), and every summand sits inside it as a clopen subspace. Everything a reader might want to know about the coproduct at this point is proved on this page.