Chapter 1

Show that the unit circle \(S^{1}\) in \(\mathbb{R}^{2}\) and the unit interval \([0,1]\) both are (Hausdorff) compactifications of \(\mathbb{R} .\) Hint: Use the fact that \(\mathbb{R}\) is homeomorphic to the open interval ] 0,1 [ and (therefore also) homeomorphic to \(\mathrm{S}^{1} \backslash\\{(1,0)\\}\).

Let \((X, \tau)\) be a topological space and consider \(X^{2}\) with the product topology. Show that \(X\) is a Hausdorff space iff the diagonal $$ \Delta=\left\\{(x, y) \in X^{2} \mid x=y\right\\} $$ is a closed subset of \(X^{2}\).

Find a continuous function \(f: \mathbb{R} \rightarrow \mathbb{R}\) that is not open.

(Topology according to Hausdorff.) Suppose that to every point \(x\) in a set \(X\) we have assigned a nonempty family \(\mathscr{U}(x)\) of subsets of \(X\) satisfying the following conditions: (i) \(x \in A\) for every \(A\) in \(\mathcal{U}(x)\). (ii) If \(A \in \mathscr{U}(x)\) and \(B \in \mathscr{U}(x)\), then there is a \(C\) in \(\mathscr{U}(x)\) with \(C \subset A \cap B\). (iii) If \(A \in \mathscr{U}(x)\), then for each \(y\) in \(A\) there is a \(B\) in \(\mathscr{\ell}(y)\) with \(B \subset A\). Show that if \(\tau\) is the weakest topology containing all \(\mathscr{U}(x), x \in X\), then \(\mathscr{U}(x)\) is a neighborhood basis for \(x\) in \(\tau\) for every \(x\) in \(X\).

Let \(f: X \rightarrow \mathbb{R}\) be a continuous function on a compact space \(X\). Show that \(f\) is bounded. Show that \(f\) attains its maximum and its minimum on \(X .\) Assume only that \(f\) is semicontinuous on \(X\) and prove "half" of the two previous assertions.

Find an open function \(f: \mathbb{R} \rightarrow \mathbb{R}\) that is not continuous. Hint: If \(x\)-int \((x)=0, \alpha_{1}, \alpha_{2}, \ldots\) is the binary expansion of the fractional part of \(x\), define \(g(x)=\lim \sup \left(n^{-1} \sum \alpha_{j} \mid j \leq n\right)\). Show that \(g(I)=[0,1]\) for every interval \(I\) in \(\mathbb{R}\). Take \(f=h \circ g\), where \(h\) is an arbitrary surjective function from \([0,1]\) to \(\mathbb{R}\).

(Topology according to Kuratowski.) Let \(\mathscr{F}(X)\) denote the system of subsets of a set \(X\), and consider a function \(Y \rightarrow \operatorname{cl}(Y)\) of \(\mathscr{F}(X)\) into itself that satisfies the four closure axioms: (i) \(\operatorname{cl}(\emptyset)=\emptyset\) (ii) \(Y \subset \operatorname{cl}(Y)\) for every \(Y\) in \(\mathscr{F}(X)\). (iii) \(\operatorname{cl}(\mathrm{cl}(Y))=\operatorname{cl}(Y)\) for every \(Y\) in \(\mathscr{S}(X)\). (iv) \(\operatorname{cl}(Y \cup Z)=\operatorname{cl}(Y) \cup \operatorname{cl}(Z)\) for all \(Y\) and \(Z\) in \(\mathscr{S}(X)\). Show that the system of sets \(F\) such that \(\mathrm{cl}(F)=F\) form the closed sets in a topology on \(X\), and that \(Y^{-}=\operatorname{cl}(Y), Y \in \mathscr{S}(X)\).

Let \(\alpha\) be a positive, irrational number and show that the relation in \(\mathbb{Z} \times \mathbb{Z}\) given by $$ \left(x_{1}, x_{2}\right) \leq\left(y_{1}, y_{2}\right) \quad \text { if } \quad \alpha\left(y_{1}-x_{1}\right) \leq y_{2}-x_{2} $$ is a total order. Sketch the set $$ \left\\{\left(x_{1}, x_{2}\right) \in \mathbb{Z} \times \mathbb{Z} \quad \mid \quad\left(x_{1}, x_{2}\right) \geq(0,0)\right\\} . $$

Let \(C\) and \(D\) be compact subsets of topological spaces \(X\) and \(Y\), respectively. Show that if \(G\) is an open subset of \(X \times Y\) containing \(C \times D\), there are open sets \(A\) and \(B\) in \(X\) and \(Y\), respectively, such that \(C \subset A, D \subset B\), and \(A \times B \subset G\).

Let \(X\) be the subset of points \((x, y)\) in \(\mathbb{R}^{2}\) such that either \(x=y=0\) or \(x y=1\), and give \(X\) the relative topology. Let \(f: X \rightarrow \mathbb{R}\) be the restriction to \(X\) of the projection of \(\mathbb{R}^{2}\) to the \(x\)-axis. Is \(f\) a continuous map? Is \(f\) an open map?

Take a bounded interval \([a, b]\) in \(\mathbb{R}\) and consider the net \(\Lambda\) of finitesubsets \(\lambda=\left\\{x_{0}, x_{1}, \ldots, x_{n}\right\\}\) of \(\mathbb{R}\) such that \(a=x_{0}

Let \(Y\) be a dense subset of a topological space \((X, \tau)\). Show that \((Y \cap A)^{-}=A^{-}\)for every open subset \(A\) of \(X .\)

An order isomorphism between two ordered sets \((X, \leq)\) and \((Y, \leq)\) is a bijective map \(\varphi: X \rightarrow Y\) such that \(x_{1} \leq x_{2}\) iff \(\varphi\left(x_{1}\right) \leq \varphi\left(x_{2}\right) .\) A segment of a well-ordered set \((X, \leq)\) is a subset of \(X\) of the form \(\min \\{x\\}\) for some \(x\) in \(X\), or \(X\) itself (the improper segment). Show that if \(X\) and \(Y\) are well-ordered sets, then either \(X\) is order isomorphic to a segment of \(Y\) (with the relative order) or \(Y\) is order isomorphic to a segment of \(X\). Hint: The system of order isomorphisms \(\varphi: X_{\varphi} \rightarrow Y_{\varphi}\) where \(X_{\varphi}\) and \(Y_{\varphi}\) are segments of \(X\) and \(Y\), respectively, is inductively ordered if we define \(\varphi \leq \psi\) to mean \(X_{\varphi} \subset X_{\psi}\) (which implies that \(\varphi=\psi \mid X_{\varphi}\) and thus \(Y_{\varphi} \subset Y_{\psi}\) ). Prove that for a maximal element \(\varphi: X_{\varphi} \rightarrow Y_{\varphi}\) either \(X_{\varphi}\) or \(Y_{\varphi}\) must be an improper segment.

(The Bohr compactification.) Show that there is a compact, abelian topological group \(\mathbb{R}\) and a continuous, in jective group homomorphism \(t: \mathbb{R} \rightarrow \widetilde{\mathbb{R}}\), such that \(t(\mathbb{R})\) is dense in \(\widetilde{\mathbb{R}}\). Hint: Let \(\mathbb{R}\) denote \(\mathbb{R}\) as a discrete group and put \(\widetilde{\mathbb{R}}=\) hom \((\mathbb{R}, \mathbb{T})\), cf. E 1.6.15. Define \(t(x)\) to be the homomorphism \(y \rightarrow \exp (\mathrm{i} x y), y \in \mathbb{R}\) for each \(x\) in \(\mathbb{R}\).

Let \(C\) and \(F\) be closed subsets of a metric space \((X, d) .\) Show that if \(C\) is compact, there is an \(x_{0}\) in \(C\) such that $$ \inf \\{d(x, y) \mid x \in C, y \in F\\}=\inf \left\\{d\left(x_{0}, y\right) \mid y \in F\right\\} . $$

Let \((X, \tau)\) be a topological space and denote by \(C(X)\) the set of continuous functions from \(X\) to \(\mathbb{R} .\) Show that the following combinations of elements \(f\) and \(g\) in \(C(X)\) again produce elements in \(C(X)\) : \(\alpha f[\) if \(\alpha \in \mathbb{R}] ;|f| ; 1 / f[\) if \(0 \notin f(X)] ; f+g ; f g ; f \vee g ; f \wedge g .\)

Show that \(\partial(Y \cup Z) \subset \partial Y \cup \partial Z\) for any two subsets \(Y\) and \(Z\) of a topological space \((X, \tau)\).

The equivalence classes of well-ordered sets modulo order isomorphism (E 1.1.3) are called ordinal numbers. Every well-ordered set has thus been assigned a "size" determined by its ordinal number. Show that the class of ordinal numbers is well-ordered. Hint: Given a collection of ordinal numbers \(\left\\{\alpha_{j} \mid j \in J\right\\}\) choose a corresponding family of well-ordered sets \(\left(X_{j} \mid j \in J\right\\}\) such that \(\alpha_{j}\) is the ordinal number for \(X_{j}\) for every \(j\) in \(J\). Now fix one \(X_{j}\). Either its equivalence class \(\alpha_{j}\) is the smallest (and we are done) or each one of the smaller \(X_{i}\) 's is order isomorphic to a proper segment \(\min \left\\{x_{i}\right\\}\) in \(X_{j}\) by \(\mathrm{E} 1.1 .3 .\) But these segments form a well-ordered set.

Let \(X\) be a Hausdorff topological space and \(Y\) be a dense subset. Assume that \(A\) is an open set and that \(A \subset B\) for some subset \(B\) such that \(B \cap Y\) is compact. Show that \(A \subset Y\). Hint: If \(x \in A \backslash Y\), there is an open neighborhood \(A_{0}\) of \(x\) disjoint from \(B \cap Y\). But \(A \cap A_{0} \cap Y \neq \emptyset\), a contradiction.

Let \(f\) and \(g\) be continuous functions between topological spaces \(X\) and \(Y\), where \(Y\) is a Hausdorff space. Show that the set $$ \\{x \in X \mid f(x)=g(x)\\} $$ is closed in \(X\). Show that if \(f\) and \(g\) are equal on a dense subset of \(X\), then \(f=g\).

Let \(\mathbb{T}\) (torus) denote the unit circle in \(\mathbb{C}\), with the relative topology. Show that the product space \(\mathbb{T}^{2}\) (the 2-torus) is homeomorphic to a closed subset of \(\mathbb{R}^{3}\). Consider now the map \(f_{\theta}: \mathbb{R} \rightarrow \mathbb{T}^{2}\) given by \(f_{\theta}(x)=(\operatorname{expix}, \exp \mathrm{i} \theta x)\) for some \(\theta\) in \(\mathbb{R}\). Prove that \(f_{\theta}\) is continuous, and find a condition on \(\theta\) that makes \(f_{\theta}\) injective.

Show that the sets \(] t, \infty[, t \in \mathbb{R}\), together with \(\emptyset\) and \(\mathbb{R}\) is a topology on \(\mathbb{R}\). Describe the closure of a point in \(\mathbb{R}\).

Let \(f: X \rightarrow Y\) and \(g: Y \rightarrow X\) be injective (but not necessarily surjective) maps between the two sets \(X\) and \(Y\). Show that there is a bijective map \(h: X \rightarrow Y\) (F. Bernstein, 1897). Hint: Define $$ A=\bigcup_{n=0}^{\infty}(g \circ f)^{n}(X \backslash g(Y)) $$ and put \(h=f\) on \(A\) and \(h=g^{-1}\) on \(X \backslash A\). Note that \(X \backslash g(Y) \subset A\), whereas \(Y \backslash f(A) \subset g^{-1}(X \backslash A)\).

(Dini's lemma.) Let \(\left(f_{\lambda}\right)_{\lambda \in A}\) be a net of real continuous functions on a compact space \(X .\) Assume that \(\lambda \leq \mu\) implies \(f_{\lambda}(x) \leq f_{\mu}(x)\) for every \(x\) in \(X\) and that there is a continuous function \(f\) on \(X\) such that \(\lim f_{\lambda}(x)=f(x)\) for every \(x\) in \(X\). Prove that \(\left(f_{\lambda}\right)\) converges uniformly to \(f\), i.e. \(\left\|f_{\lambda}-f\right\|_{\infty} \rightarrow 0\).

A topological group is a group \(G\) equipped with a topology \(\tau\) such that the map \((x, y) \rightarrow x^{-1} y\) from \(G \times G\) (with the product topology) into \(G\) is continuous. Show in this case that the maps \(x \rightarrow x^{-1}\), \(x \rightarrow x y\), and \(x \rightarrow y x\) are homeomorphisms of \(G\) onto itself for every fixed \(y\) in \(G\). Show that \(O(x)=x \mathcal{O}(e)=\mathcal{O}(e) x\) for every \(x\) in \(G\), where \(e\) denotes the unit of \(G\). Show that \(\mathcal{O}(e)\) has a basis consisting of symmetric sets \(\left(A^{-1}=A\right)\). Show that a (group) homomorphism \(\pi: G \rightarrow H\) between topological groups \(G\) and \(H\) is continuous iff \(\pi\) is continuous at \(e\). Show that a topological group \(G\) is a Hausdorff space if for every \(x\) in \(G\) with \(x \neq e\) there is either an \(A\) in \(\mathcal{O}(x)\) such that \(e \notin A\) or a \(B\) in \(\mathcal{O}(e)\) such that \(x \notin B\). Show that if \(A \in \tau\), then \(A B \in \tau\) and \(B A \in \tau\) for every subset \(B\) of \(G\). Show that every open subgroup of \(G\) is closed.

Let \(X, Y\), and \(Z\) be topological spaces and give \(X \times Y\) the product topology. Show that if a function \(f: X \times Y \rightarrow Z\) is continuous, then it is separately continuous in each variable [i.e., for each \(x\) in \(X\) the function \(y \rightarrow f(x, y)\) is continuous from \(Y\), to \(Z\), and similarly for each \(y\) in \(Y\) ]. Show by an example that the converse does not hold. Hint: Try \(f(x, y)=x y\left(x^{2}+y^{2}\right)^{-1 / 2}\) if \((x, y) \neq(0,0)\) and \(f(0,0)=0\).

Let \(\left(x_{\lambda}\right)_{\lambda \in \Lambda}\) be a net in a set \(X\). Show that the system \(\mathscr{F}\) of subsets \(A\) of \(X\), such that the net is in \(A\) eventually, is a filter (E 1.3.4). Show that the net converges to a point \(x\) in \(X\) iff the filter converges to \(x\).

We define an equivalence relation on the class of sets by setting \(X \sim Y\) if there exists a bijective map \(h: X \rightarrow Y\). Each equivalence class is called a cardinal number. Show that the natural numbers are the cardinal numbers for finite sets. Discuss the "cardinality" of some infinite sets, e.g. \(\mathbb{N}, \mathbb{Z}, \mathbb{R}\), and \(\mathbb{R}^{2}\).

Let \(X\) be an infinite set and let \(\tau\) denote the system of subsets \(A\) of \(X\) such that \(X \backslash A\) is finite, together with the set \(\emptyset .\) Show that \(\tau\) is a topology on \(X\), and that \((X, \tau)\) is compact, but not a Hausdorff space. Show that points are closed sets in \(X\). Show that every infinite subset of \(X\) is dense and deduce that \((X, \tau)\) is separable. Assume that \(X\) is uncountable and show that \((X, \tau)\) satisfies neither the first nor the second axiom of countability.

(The Sorgenfrey line.) Give the set \(\mathbb{R}\) the topology \(\tau\) for which a basis consists of the half-open intervals \([y, z[\), where \(y\) and \(z\) range over R. Show that every basis set is closed in \(\tau\). Show that \((\mathbb{R}, \tau)\) is a separable space that satisfies the first but not the second axiom of countability. Hint: If \(\rho\) is some basis for \(\tau\) and \(x \in \mathbb{R}\), then \(\rho\) must contain a set \(A\) such that \(x=\operatorname{Inf}\\{y \in A\\} .\)

Fix a prime number \(p\), and take the sets $$ A(n, \alpha)=\left\\{m \in \mathbb{Z} \mid m=n+q p^{\alpha}, q \in \mathbb{Z}\right\\} $$ where \(n \in \mathbb{Z}\) and \(\alpha \in \mathbb{N} \cup\\{0\\}\), to be the basis for a topology \(\tau\) on \(\mathbb{Z}\). Show that \(\tau\) is induced by the metric \(d\) given by \(d(n, m)=p^{-\alpha}\), where \(\alpha\) is the largest number (in \(\mathbb{N} \cup\\{0\\}\) ) such that \(p^{\alpha}\) divides \(|n-m| .\) Show in particular that \(A(n, \alpha)=\left\\{m \in \mathbb{Z} \mid d(n, m) \leq p^{-\alpha}\right\\} .\) Show that \((\mathbb{Z}, \tau)\) has no isolated points and that the space is not locally compact.

(The Cantor set.) Let \(C\) denote the set of real numbers \(x\) of the form $$ x=\sum_{n=1}^{\infty} \alpha_{n} 3^{-n}, \quad \text { where } \alpha_{n}=0 \text { or } \alpha_{n}=2 $$ Show that \(C=\bigcap C_{n}\), where \(C_{1}=[0,1]\) and where \(C_{n+1}\) is obtained from \(C_{n}\) by deleting the (open) middle third of each of the intervals that belong to \(C_{n}\). Deduce from this that \(C\) (in the relative topology induced from \(\mathbb{R}\) ) is a compact Hausdorff space. Show that \(C\) as a subset of \(\mathbb{R}\) has empty interior, but no isolated points. Show that every \(x\) in \(C\) has a unique expression \(x=\sum \alpha_{n} 3^{-n}\) and that \(C\) is homeomorphic with the product space \(\\{0,1\\}^{N}\) (and thus uncountable). Show that the map \(f: C \rightarrow[0,1]\) given by \(f(x)=\sum \alpha_{n} 2^{-n-1}\), where \(x=\sum \alpha_{n} 3^{-n}\), is continuous and surjective.

(The Sorgenfrey plane.) Give the set \(\mathbb{R}^{2}\) the topology \(\tau^{2}\), for which a basis consist of products of half-open intervals \(\left[y_{1}, z_{1}\left[\times\left[y_{2}, z_{2}[\right.\right.\right.\), where \(y_{1}, y_{2}, z_{1}\), and \(z_{2}\) range over \(\mathbb{R}\). Show that \(\left(\mathbb{R}^{2}, \tau^{2}\right)\) is a separable space. Show that the subset \(\left\\{(x, y) \in \mathbb{R}^{2} \mid x+y=0\right\\}\) is discrete in the relative topology (and thus nonseparable), but closed in \(\mathbb{R}^{2}\).

A set is called countable (or countably infinite) if it has the same cardinality (cf. E 1.1.6) as the set No natural numbers. Show that there is a well-ordered set \((X, \leq)\), which is itself uncountable, but which has the property that each segment min \(\\{x\\}\) is countable if \(x \in X\). Hint: Choose a well-ordered set \((Y, \leq)\) that is uncountable. The subset \(Z\) of elements \(z\) in \(Y\) such that the segment \(\min \\{z\\}\) is uncountable is either empty (and we are done) or else has a first element \(\Omega\) Set \(X=\min \\{\Omega\\}\). The ordinal number (corresponding to) \(\Omega\) is called the first uncountable ordinal.

A subset \(Y\) of a topological space \((X, \tau)\) is an \(F_{\sigma}\)-set, if \(Y=\bigcup F_{n}\), where \(\left(F_{n}\right)\) is a sequence of closed subsets of \(X\). Show that every \(F_{\sigma}\)-set in a normal space is normal (in the relative topology). Hint: Assume first that \(G\) is an open \(F_{\sigma}\)-set of the form \(G=\bigcup G_{n}\), where each \(G_{n}\) is open and \(G_{n}^{-} \subset G_{n+1} .\) Let \(f: C \rightarrow[0,1]\) be a continuous function on a relatively closed subset \(C\) of \(G\). Construct inductively continuous functions \(f_{n}\) on \(\left(G_{n}^{-} \cap G\right) \cup G_{n-1}^{-}\)such that \(f_{n}\left|G_{n-1}^{-}=f_{n-1}\right| G_{n-1}^{-}\)and \(f_{n}\left|G_{n}^{-} \cap C=f\right| G_{n}^{-} \cap C\). Then \(\left(f_{n}\right)\) defines a unique, continuous extension of \(f\), which proves that \(G\) is normal. In the general case \(Y=\bigcup F_{n}\), let \(E\) and \(F\) be closed subsets of \(X\) such that \((E \cap Y) \cap(F \cap Y)=\emptyset\). Thus, \(H=X \backslash(E \cap F)\) is open and \(Y \subset H\). Construct inductively open subsets \(G_{n}\) such that \(F_{n} \bigcup G_{n}^{-} \subset\) \(G_{n+1} \subset H\). Then \(G=\bigcup G_{n}\) is open and normal, so \(E \cap G\) and \(F \cap G\) can be separated by open, disjoints subsets \(A\) and \(B\) in \(G\). Consequently, \(A \cap Y\) and \(B \cap Y\) are relatively open subsets of \(Y\) that separate \(E \cap Y\) and \(F \cap Y\).

Let \(\mathfrak{X}\) be a vector space over a field \(\mathbb{F} .\) A basis for \(\mathfrak{X}\) is a subset \(\mathfrak{B}=\left\\{e_{j} \mid j \in J\right\\}\) of linearly independent vectors from \(\mathfrak{X}\), such that every \(x\) in \(\mathfrak{X}\) has a (necessarily unique) decomposition as a finite linear combination of vectors from \(\mathfrak{B}\). Show that every vector space has a basis. Hint: A basis is a maximal element in the system of linearly independent subsets of \(\mathfrak{X}\).

(Completely normal spaces.) Show that the following conditions on a topological space \((X, \tau)\) are equivalent: (i) Every subset of \(X\) (equipped with the relative topology) is a normal space. (ii) Every open subset of \(X\) is a normal space. (iii) If \(Y\) and \(Z\) are subsets of \(X\) such that \(Y^{-} \cap Z=Y \cap Z^{-}=\emptyset\), they can be separated by open, disjoint sets in \(X\). Hints: For (ii) \(\Rightarrow\) (iii), consider the open, hence normal, subspace \(G=\) \(X \backslash\left(Y^{-} \cap Z^{-}\right)\)and separate \(Y^{-} \cap G\) and \(Z^{-} \cap G\) with open, disjoint sets \(A\) and \(B\). Show that, in fact, \(Y \subset A\) and \(Z \subset B\). For (iii) \(\Rightarrow\) (i) consider a pair \(Y, Z\) of relatively closed, dis joint subsets of an arbitrary subset \(H\) of \(X\). Show that \(Y^{-} \cap Z=Z \cap Y^{-}=\emptyset\).

On \(\mathbb{R}\) we consider the equivalence relation \(\sim\) given by \(x \sim y\) if \(x-y \in \mathbb{Z}\). Describe the quotient space and the quotient topology.

A topological space \((X, \tau)\) is a Lindelöf space if each family \(\sigma\) in \(\tau\) that covers \(X\) (i.e. \(X=\bigcup A, A \in \sigma\) ) contains a countable subset \(\left\\{A_{n} \mid n \in \mathbb{N}\right\\} \subset \sigma\) that covers \(X\). Show that \((X, \tau)\) is a Lindelöf space if \(\tau\) satisfies the second axiom of countability. Hint: If \(\sigma\) is an open covering of \(X\) and \(\left\\{B_{n} \mid n \in \mathbb{N}\right\\}\) is a basis for \(\tau\), then there is a countable subset \(\left\\{B_{n_{k}} \mid k \in \mathbb{N}\right\\}\) of basis sets such that each \(B_{n_{k}}\) is contained in some \(A_{k}\) from \(\sigma .\) But this subset must cover \(X\).

Show that there exists a discontinuous function \(f: \mathbb{R} \rightarrow \mathbb{R}\), such that \(f(x+y)=f(x)+f(y)\) for all real numbers \(x\) and \(y\). Show that \(f(I)\) contains arbitrarily (numerically) large numbers for every (small) interval \(I\) in \(\mathbb{R}\). Hint: Let Q denote the field of rational numbers and apply E 1.1.9 with \(X=\mathbb{R}\) and \(F=\mathbb{Q}\) to obtain what is called a Hamel basis for \(\mathbb{R}\). Show that \(f\) can be assigned arbitrary values on the Hamel basis and still have an (unique) extension to an additive function on \(\mathbb{R}\).

(Topological direct sum.) Let \(\left(X_{1}, \tau_{1}\right)\) and \(\left(X_{2}, \tau_{2}\right)\) be topological spaces and let \(X\) denote the disjoint union of \(X_{1}\) and \(X_{2} .\) Find the topology \(\tau\) on \(X\) that contains \(X_{1}\) and \(X_{2}\) and for which the relative topology on \(X_{j}\) is \(\tau_{j}\) for \(j=1\), 2. Show that \(\tau\) is the final topology corresponding to the embedding maps \(l_{j}: X_{j} \rightarrow X\) for \(j=1,2\).

Let \(X\) be a set and \(\mathscr{S}(X)\) the family of all subsets of \(X\). Show that the cardinality of the set \(\mathscr{S}(X)\) is strictly larger than that of \(X\), cf. E 1.1.6. Hint: If \(f: X \rightarrow \mathscr{S}(X)\) is a bijective function, set $$ A=\\{x \in X \mid x \notin f(x)\\} $$ and take \(y=f^{-1}(A)\). Either possibility \(y \in A\) or \(y \notin A\) will lead to a contradiction.

(Inductive limits.) Let \(\left(X_{n}, \tau_{n}\right)\) be a sequence of topological spaces and assume that there is a continuous injective map \(f_{n}: X_{n} \rightarrow X_{n+1}\) for every \(n\). Identifying every \(X_{n}\) with a subset of \(X_{n+1}\) (equipped with the relative topology if \(f_{n}\) is a homeomorphism on its image), we form \(X=\bigcup X_{n}\), and give it the final topology induced by the maps \(f_{n}: X_{n} \rightarrow X\). Take \(X_{n}=\mathbb{R}^{n}\) with the natural embeddings \(f_{n}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} \times \mathbb{R}\), so that \(X=\mathbb{R}^{N}\). Show that the inductive limit topology on \(\mathbb{R}^{\mathbb{N}}\) is stronger than the product topology. Hint: Show that the cube (] \(0,1[)^{N}\) is open in the inductive limit topology on \(\mathbb{R}^{\mathbb{N}}\).

Show that a paracompact Hausdorff space is normal. Hint: If \(E\) and \(F\) are disjoint, closed subsets of \(X\), use regularity to cover \(E\) with a family \(\left\\{A_{j} \mid j \in J\right\\}\) of open sets such that \(A_{j}^{-} \cap F=\emptyset\). Use paracompactness to conclude that the covering may be taken to be locally finite. Set $$ A=\bigcup A_{j}, \quad B=X \backslash \bigcup A_{j}^{-}, \quad B^{o}=X \backslash\left(\bigcup A_{j}^{-}\right)^{-} $$ Show that \(E \subset A, F \subset B\) and \(A \cap B=\emptyset\). Use the local finiteness to conclude that \(B=B^{\circ}\).

(Connected spaces.) A topological space \((X, \tau)\) is connected if it cannot be decomposed as a union of two nonempty disjoint open sets. A subset of \(X\) is clopen if it is both open and closed. Show that \(X\) is connected iff \(\emptyset\) and \(X\) are the only clopen subsets. Let \(f: X \rightarrow Y\) be a surjective continuous map between topological spaces. Show that \(Y\) is connected if \(X\) is.

(Arcwise connected spaces.) A topological space \((X, \tau)\) is arcwise connected if for every pair \(x, y\) in \(X\) there is a continuous function \(f:[0,1] \rightarrow X\) such that \(f(0)=x\) and \(f(1)=y\). Geometrically speaking, \(f([0,1])\) is the curve or arc that joins \(x\) to \(y\). Show that an arcwise connected space is connected (E 1.4.13). Show that \(Y\) is arcwise connected if it is the continuous image of an arcwise connected space (cf. E 1.4.13).

Show that \(\mathbb{R}\) is not homeomorphic to \(\mathbb{R}^{2}\). Hint: \(\mathbb{R}^{2} \backslash\left\\{x_{1}, x_{2}\right\\}\) is a connected space, but \(\mathbb{R} \backslash\\{x\\}\) is disconnected.

Let \(E\) be a connected subset of \(\mathbb{R}\). Show that \(E\) is an interval (possibly unbounded).

(Homotopy.) Two continuous maps \(f: X \rightarrow Y\) and \(g: X \rightarrow Y\) between topological spaces \(X\) and \(Y\) are homotopic if there is a continuous function \(F:[0,1] \times X \rightarrow Y\) (where \([0,1] \times X\) has the product topology), such that \(F(0, x)=f(x)\) and \(F(1, x)=g(x)\) for every \(x\) in \(X\). Intuitively speaking, the homotopy \(F\) represents a continuous deformation of \(f\) into \(g\). Show that any continuous function \(f: \mathbb{R}^{n} \rightarrow Y\) is homotopic to a constant function, and that the same is true for any continuous function \(g: X \rightarrow \mathbb{R}^{n}\). Show that the identity function r. \(\mathrm{S}^{1} \rightarrow \mathrm{S}^{1}\) [where \(\mathrm{S}^{1}=\left\\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2}=1\right\\}\) ] is not homotopic to a constant function.

(Contractible spaces.) A topological space \((X, \tau)\) is contractible if it is homotopic to a point (E 1.4.20). Show that \(X\) is contractible iff the identity map \({ }_{X}\) is homotopic to a constant map. Show that every convex subset of \(\mathbb{R}^{n}\) is contractible.