Quantum Homotopy Type Theory

Definition Let \(X\) be a topological space. Define an equivalence relation \(\sim\) on \(X\) by: \(x \sim y\) iff there exists a connected subspace \(C\) of \(X\) such that \(x \in C\) and \(y \in C\). The (connected) components of \(X\) are the equivalence classes of \(X\) under this equivalence relation. Theorem 25.1 The components of \(X\) are the maximal connected subspaces of \(X\). Definition Let \(X\) be a topological space. Define an equivalence relation \(\sim\) on \(X\) by: \(x \sim y\) iff there exists a path in \(X\) from \(x\) to \(y\). The path components of \(X\) are the equivalence classes of \(X\) under this equivalence relation. Theorem 25.2 The path components of \(X\) are the maximal path connected subspaces of \(X\). Proposition Every component is closed. Proof Since the closure of a connected subspace is connected. Definition Let \(X\) be a topological space. For \(x \in X\), \(X\) is locally (path) connected at \(x\) iff, for ever...

Definition A linear continuum is a linearly ordered set with more than one element that is dense and complete. Proposition A convex subset of a linear continuum that has more than one element is a linear continuum. Theorem 24.1 A linear continuum under the order topology is connected. Proof Let \(L\) be a linear continuum. Assume for a contradiction \(U\) and \(V\) form a separation of \(L\). Pick \(a \in U\) and \(b \in V\). Assume w.l.o.g. \(a Let \(c = \sup (U \cap [a,b])\) Case \(c \in U\) Pick \(e > c\) such that \([c,e) \subseteq U\). \(e \leq b\) Pick \(f\) such that \(c \(f \in U\) This contradicts the fact that \(c\) is an upper bound for \(U \cap [a,b]\). Case \(c \in V\) Pick \(e \(e\) is an upper bound for \(U \cap [a,b]\). Let \(x \in U \cap [a,b]\) \(x \leq c\) \(x \notin (e,c]\) \(x \leq e\) This contradicts the leastness of \(c\). \(\Box...

Definition Let \(X\) be a topological space. A separation of \(X\) is a pair of disjoint nonempty open sets \(U\), \(V\) such that \(U \cup V = X\). The space \(X\) is connected iff there does not exist a separation of \(X\). Lemma 23.1 Let \(X\) be a topological space. Let \(Y\) be a subspace of \(X\). A separation of \(Y\) is a pair of disjoint nonempty sets \(A\) and \(B\) whose union is \(Y\), neither of which contains a limit point (in \(X\)) of the other. Example 4 The rationals are not connected. \(\{q \in \mathbb{Q} \mid q^2 2\}\) form a separation. Lemma 23.2 Let \(X\) be a topological space. If \(U\) and \(V\) form a separation of \(X\), and \(Y\) is a connected subspace of \(X\), then \(Y \subseteq U\) or \(Y \subseteq V\). Theorem 23.3 Let \(X\) be a topological space. The union of a set of connected subspaces of \(X\) that have a point in common is connected. Theorem 23.4 Let \(X\) be a topological space. Let \(A\) be a connected subspace of \...

Definition A topological group \(G\) consists of a group \(G\) and a \(T_1\) topology on \(G\) such that the multiplication of \(G\) and the function mapping \(x\) to \(x^{-1}\) are continuous. Exercise 1 Let \(G\) be a group with a \(T_1\) topology. Then \(G\) is a topological group if and only if the function \(G^2 \rightarrow G\) that maps \((x,y) to xy^{-1}\) is continuous. Exercise 2 The following are topological groups: \((\mathbb{Z}, +)\), \((\mathbb{R}, +)\), \((\mathbb{R}_+, \cdot)\), \((\{z \in \mathbb{C} \mid |z| = 1\}, \cdot)\), \(GL_n(\mathbb{R}) \subseteq \mathbb{R}^{n^2}\). Exercise 3 Let \(G\) be a topological group. Let \(H\) be a subspace of \(G\) that is also a subgroup of \(G\). Then \(H\) and \(\overline{H}\) are topological groups. Solution \(H\) is a topological group. \(H\) is \(T_1\) Any subspace of a \(T_1\) space is \(T_1\). The function mapping \((x,y)\) to \(xy^{-1}\) is continuous. It is th...

Definition Let \(X\) and \(Y\) be topological spaces. Let \(p : X \rightarrow Y\). Then \(p\) is strongly continuous iff, for all \(U \subseteq Y\), we have \(U\) is open in \(Y\) if and only if \(p^{-1}(U)\) is open in \(X\). Definition Let \(X\) and \(Y\) be topological spaces. Let \(p : X \rightarrow Y\). Then \(p\) is a quotient map iff \(p\) is surjective and strongly continuous. Definition Let \(X\) and \(Y\) be sets. Let \(p : X \rightarrow Y\). Let \(C \subseteq X\). Then \(C\) is saturated iff, for all \(x,y \in X\), if \(x \in C\) and \(p(x) = p(y)\) then \(y \in C\). Proposition Let \(X\) and \(Y\) be topological spaces. Let \(p : X \rightarrow Y\) be surjective. Then the following are equivalent. \(p\) is a quotient map. \(p\) is continuous and maps saturated open sets to open sets. \(p\) is continuous and maps saturated closed sets to closed sets. Definition Let \(X\) and \(Y\) be topological spaces. Let \(f : X \rightarrow Y\). Then \(f...

Proposition Let \((X,d)\) be a metric space and \(Y \subseteq X\). Then \(d \restriction Y^2\) is a metric on \(Y\) that induces the subspace topology. Proposition Every metric space is Hausdorff. Proposition Let \(((X_n, d_n))\) be a sequence of metric spaces. Define \(D\) on \(prod_n X_n\) by \(D((x_n)) = \sup_n (\overline{d_n}(x_n, y_n)/n)\), where \(\overline{d_n}\) is the standard bounded metric corresponding to \(d_n\). Then \(D\) is a metric on \(\prod_n X_n\) that induces the product topology. Theorem 21.1 Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces. Let \(f : X \rightarrow Y\). Then \(f\) is continuous if and only if, for all \(x \in X\) and \(\epsilon > 0\), there exists \(\delta > 0\) such that, for all \(y \in X\), \[ d_X(x,y) Definition Let \(X\) be a topological space. Let \(x \in X\). A basis at the point \(x\) is a set \(\mathcal{B}\) of neighbourhoods of \(x\) such that every neighbourhood of \(x\) includes a member of \(\mathcal{B}\)...

Definition A metric on a set \(X\) is a function \(d : X^2 \rightarrow \mathbb{R} \) such that, for all \(x,y,z \in X\): \(d(x,y) \geq 0\) \(d(x,y) = 0\) iff \(x = y\) \(d(x,y) = d(y,x)\) \em{Triangle Inequality} \(d(x,z) \leq d(x,y) + d(y,z)\) We call \(d(x,y)\) the distance between \(x\) and \(y\). A metric space is a pair \((X,d)\) where \(X\) is a set and \(d\) is a metric on \(X\). We usually write \(X\) for the metric space \((X,d)\). Definition Let \(X\) be a metric space. Let \(x \in X\) and \(\epsilon > 0\). The open ball with centre \(x\) and radius \(\epsilon\) is \[ B_d(x,\epsilon) = \{y \in X \mid d(x,y) Definition Let \(X\) be a metric space. The metric topology on \(X\) is the topology generated by the basis consisting of all the open balls. Proposition Let \(X\) be a metric space and \(U \subseteq X\). Then \(U\) is open if and only if, for all \(x \in U\), there exists \(\epsilon > 0\) such that \(B(x,\epsilon) \subseteq...

Definition Let \(\{X_i\}_{i \in I}\) be a family of topological spaces. The box topology on \(\prod_{i \in I} X_i\) is the topology generated by the basis \(\{\prod_{i \in I} U_i \mid \forall i \in I. U_i \text{ is open in } X_i\}\). Definition Let \(\{X_i\}_{i \in I}\) be a family of topological spaces. The product topology on \(\prod_{i \in I} X_i\) is the topology generated by the subbasis \(\{\pi_i^{-1}(U) \mid i \in I, U \text{ is open in } X_i\}\). In this topology, \(\prod_{i \in I} X_i\) is called a product space . If we write just \(\prod_{i \in I} X_i\), it is to be understood that this set is given the product topology. Theorem 19.1 Let \(\{X_i\}_{i \in I}\) be a family of topological spaces. A basis for the product topology on \(\prod_{i \in I} X_i\) is given by the set of all sets of the form \(\prod_{i \in I} U_i\) where each \(U_i\) is open in \(X_i\), and \(U_i = X_i\) for all but finitely many \(i \in I\). Theorem 19.2 Let \(\{X_i\}_{i \in I}\) be a ...

Continuity of a Function Definition Let \(X\) and \(Y\) be topological spaces. Let \(f : X \rightarrow Y\). Then \(f\) is continuous iff, for every open set \(V\) in \(Y\), the set \(f^{-1}(V)\) is open in \(X\). Proposition Let \(X\) and \(Y\) be topological spaces. Let \(f : X \rightarrow Y\). Let \(\mathcal{S}\) be a subbasis for the topology on \(Y\). Then \(f\) is continuous iff, for every \(U \in \mathcal{S}\), we have \(f^{-1}(U)\) is open in \(X\). Definition Let \(X\) and \(Y\) be topological spaces. Let \(f : X \rightarrow Y\). Let \(x \in X\). Then \(f\) is continuous at \(x\) iff, for every neighbourhood \(V\) of \(f(x)\), there exists a neighbourhood \(U\) of \(x\) such that \(f(U) \subseteq V\). Theorem 18.1 Let \(X\) and \(Y\) be topological spaces. Let \(f : X \rightarrow Y\). The following are equivalent. \(f\) is continuous. For every \(A \subseteq X\), we have \(f(\overline{A}) \subseteq \overline{f(A)}\). For every closed set \(B\) in...

Closed Sets Definition Let \(X\) be a topological space and \(C \subseteq X\). Then \(C\) is closed iff \(X - C\) is open. Theorem 17.1 Let \(X\) be a topological space. \(\emptyset\) and \(X\) are closed. The intersection of a nonempty set of closed sets is closed. The union of two closed sets is closed. Theorem 17.2 Let \(X\) be a topological space. Let \(Y\) be a subspace of \(X\) and \(A \subseteq Y\). Then \(A\) is closed in \(Y\) iff there exists a closed set \(C\) in \(X\) such that \(A = C \cap Y\). Theorem 17.3 Let \(X\) be a topological space. Let \(Y\) be a subspace of \(X\) and \(A \subseteq Y\). If \(A\) is closed in \(Y\) and \(Y\) is closed in \(X\) then \(A\) is closed in \(X\). Closure and Interior of a Set Definition Let \(X\) be a topological space. Let \(A \subseteq X\). The interior of \(A\), denoted \(A^\circ\), is the union of all the open sets included in \(A\). Definition Let \(X\) be a topological space. Let \(A \subs...

Definition Let \(X\) be a topological space. Let \(Y\) be a subset of \(X\). The subspace topology on \(Y\) is \(\{ U \cap Y \mid U \text{ open in } X \}\). We call \(Y\) with this topology a subspace of \(X\). Lemma 16.1 Let \(X\) be a topological space. Let \(Y\) be a subspace of \(X\). Let \(\mathcal{B}\) be a basis for the topology on \(X\). Then \(\{B \cap Y \mid B \in \mathcal{B}\}\) is a basis for the subspace topology on \(Y\). Lemma 16.2 Let \(X\) be a topological space. Let \(Y\) be a subspace of \(X\). Let \(U \subseteq Y\). If \(U\) is open in \(Y\) and \(Y\) is open in \(X\) then \(U\) is open in \(X\). Theorem 16.3 Let \(X\) and \(Y\) be topological spaces. Let \(A\) be a subspace of \(X\) and \(B\) a subspace of \(Y\). Then the product topology on \(A \times B\) is the same as the topology \(A \times B\) inherits as a subspace of \(X \times Y\). Example 2 It is not true in general that, if \(X\) is a linearly ordered set with the order topology and...

Definition Let \(X\) and \(Y\) be topological spaces. The product topology on \(X \times Y\) is the topology generated by the basis consisting of all sets of the form \(U \times V\) where \(U\) is open in \(X\) and \(V\) is open in \(Y\). Theorem 15.1 Let \(X\) and \(Y\) be topological spaces. Let \(\mathcal{B}\) be a basis for the topology on \(X\) and \(\mathcal{C}\) a basis for the topology on \(Y\). Then \(\{ B \times C \mid B \in \mathcal{B}, C \in \mathcal{C} \}\) is a basis for the product topology on \(X \times Y\). Example 1 The standard topology on \(\mathbb{R}^2\) is the product topology. Theorem 15.2 The set \(\{\pi_1^{-1}(U) \mid U \text{ open in } X \} \cup \{\pi_2^{-1}(V) \mid V \text{ open in } Y \}\) is a subbasis for the product topology on \(X \times Y\).

Definition Let \(X\) be a linearly ordered set with more than one element. The order topology on \(X\) is the topology generated by the basis consisting of: All open intervals All intervals of the form \([\bot, b)\) where \(\bot\) is the smallest element of \(X\), if this exists. All intervals of the form \((a, \top]\) where \(\top\) is the greatest element of \(X\), if this exists. Example 1 The standard topology on \(\mathbb{R}\) is the order topology. The order topology is the topology generated by the subbasis of all open rays.

Definition Let \(X\) be a set. A basis for a topology on \(X\) is a set \(\mathcal{B} \subseteq \mathcal{P} X\) such that: \( \bigcup \mathcal{B} = X\) For all \(B_1, B_2 \in \mathcal{B}\) and \(x \in B_1 \cap B_2\), there exists \(B_3 \in \mathcal{B}\) such that \(x \in B_3 \subseteq B_1 \cap B_2\). The topology generated by \(\mathcal{B}\) is then \( \{U \in \mathcal{P} X \mid \forall x \in U. \exists B \in \mathcal{B}. x \in B \subseteq U \}\). This is the same as \( \{ \bigcup \mathcal{B}_0 \mid \mathcal{B}_0 \subseteq \mathcal{B} \}\). Lemma 13.2 Let \(X\) be a topological space. Let \(\mathcal{B}\) be a set of open sets of \(X\) such that, for every open set \(U\) and each \(x \in U\), there exists \(B \in \mathcal{B}\) such that \(x \in B \subseteq U\). Then \(\mathcal{B}\) is a basis for the topology on \(X\). Lemma 13.3 Let \(\mathcal{B}\) and \(\mathcal{B}'\) be bases for the topologies \(\mathcal{T}\) and \(\mathcal{T}'\) respectively. Then ...

I'm going to speed up, otherwise we'll never get to homotopy type theory. I'm not going to write out proofs if I can do them in my head, and I'll only write solutions to exercises here if I cannot do them in my head. Definition: Topology A topology on a set \(X\) is a set \(\mathcal{T}\) of subsets of \(X\), whose elements are called open sets , such that: \(X \in \mathcal{T}\) The union of any subset of \(\mathcal{T}\) is in \(\mathcal{T}\). The intersection of two elements of \(\mathcal{T}\) is in \(\mathcal{T}\). A topological space is a pair \((X, \mathcal{T})\) such that \(X\) is a set and \(\mathcal{T}\) is a topology on \(X\). Example 2 For any set \(X\), the discrete topology on \(X\) is \(\mathcal{P} X\). The indiscrete topology or trivial topology on \(X\) is \(\{\emptyset, X\}\). Example 3 For any set \(X\), the finite complement topology on \(X\) is \(\{U \in \mathcal{P} X \mid X - U \text{ is finite}\} \cup \{\emptyset\}\)...

Making a list here of things that come up in references that I'd like to read: R. M. Smullyan. The continuum hypothesis. In The Mathematical Sciences, A Collection of Essays . 1969. P. J. Campbell. The origin of “Zorn’s lemma”. Historia Mathematica , 5:77–89, 1978 G. H. Moore. Zermelo’s Axiom of Choice . Springer-Verlag, New York, 1982

Definition: Countably Infinite A set is countably infinite iff it is bijective with \(\mathbb{Z}_+\). Example The set \(\mathbb{Z}\) is countably infinite. Definition: Countable A set is countable iff it is either finite or countably infinite; otherwise it is uncountable . Principle of Recursive Definition Let \(A\) be a set. Let \(a_0 \in A\). Let \(\rho\) be a function from \( \{ f \mid \exists n \in \mathbb{Z}_+. f : \{1, \ldots, n\} \rightarrow A \} \) to \(A\). Then there exists a unique function \(h : \mathbb{Z}_+ \rightarrow A\) such that \[ \begin{align} h(1) & = a_0 \\ h(i) & = \rho(h \restriction \{1, \ldots, i-1\}) & (i > 1) \end{align} \] Proof For every positive integer \(n\), there exists a unique function \(f_n : \{1, \ldots, n\} \rightarrow A\) such that \(f_n(1) = a_0\) and, for all \(i = 2, 3, \ldots, n\), we have \(f_n(i) = \rho(f_n \restriction \{1, \ldots, i-1\})\). Let \(P(n)\) be the statement: there exists a uniq...

Definition: Finite A set \(A\) is finite iff it is bijective with \(S_n\) for some positive integer \(n\), in which case we say its cardinality is \(n-1\). Lemma 6.1 Let \(n\) be a positive integer. Let \(A\) be a set. Let \(a_0 \in A\). The set \(A\) is bijective with \(S_{n+2}\) if and only if \(A - \{a_0\}\) is bijective with \(S_{n+1}\). Proof If \(A\) is bijective with \(S_{n+2}\) then \(A - \{a_0\}\) is bijective with \(S_{n+1}\). Assume \(f : A \rightarrow S_{n+2}\) is a bijection. Define \(g : A - \{a_0\} \rightarrow S_{n+1}\) by \(g(x) = f(x)\) if \(f(x) Let \(x \in A - \{a_0\}\), and assume \(f(x) = n+1\). Then \(f(a_0) \(g\) is injective. Let \(x,y \in A - \{a_0\}\). Assume \(g(x) = g(y)\). Prove: \(x = y\). Case \(f(x) We have \(f(x) = g(x) = g(y) = f(y)\) and so \(x = y\) since \(f\) is injective. Case \(f(x) We have \(f(x) = g(x) = g(y) = f(a_0)\), contrad...

Definition: Family Let \(A\) be a set. A family of elements of \(A\) is a function \(a\) with codomain \(A\). We call its domain \(I\) the index set , and we write \(a_i\) for the value of \(a\) at \(i \in I\). We write the family as \(\{a_i\}_{i \in I}\), and call it a family indexed by \(I\). A sequence is a family indexed by \(\mathbb{Z}_+\). Definition: Euclidean \(m\)-space For \(m \in \mathbb{Z}_+\), we call \(\mathbb{R}^m\) Euclidean \(m\)-space . Exercises Exercise 1 Show that there is a bijective correspondence of \(A \times B\) with \(B \times A\). Solution The function mapping \((a,b)\) to \((b,a)\) is such a correspondence. Exercise 2 (a) Show that if \(n > 1\) there is a bijective correspondence of \[ A_1 \times \cdots \times A_n \text{ with } (A_1 \times \cdots \times A_{n-1}) \times A_n \enspace . \] Solution The function mapping \((a_1, \ldots, a_n)\) to \(((a_1, \ldots, a_{n-1}), a_n)\) is such a correspondence. (b) Given the i...