Topology - Winter 2024/25

Sep 19, 2024 Kamillo Ferry Topology Exercises

I am a teaching assistant for the BMS basic course on general topology taught in Winter 2024/25. The exercise sessions take place on Thursday 10:00 in MA742.

There will be an exercise sheet every week. Solutions to the exercises may be handed in up until the exercise session via e-mail to ferry@math.tu-berlin.de. There is also the possibility of presenting the solution to an exercise during the exercise session.

Exercise sheet 0

  1. Let ${\tau_i}$ be a collection of topologies on a set $X$ indexed by $i\in I$. Show that the intersection $\tau\coloneqq\bigcap_{i\in I}\tau_i$ is a topology.
  2. Show that if $\mathcal{B}$ is a basis for a topology $\tau$, then $\tau$ is equal to the intersection of all topologies on $X$ containing $\mathcal{B}$. Does the same hold for a subbasis?
  3. Let $(X,\tau)$ be a topological space. Show that $(X,\tau)$ is Fréchet if and only if every point $x\in X$ is closed.
  4. Let $\mathcal{B}$ and $\mathcal{B}'$ be bases for topologies $\tau$ and $\tau'$ respectively. Prove the equivalence of the following statements.
    1. $\tau'$ is finer than $\tau$.
    2. For each $x\in X$ and basis element $B\in\mathcal{B}$ containing $x$, there is a basis element $B'\in\mathcal{B}'$ such that $x\in B'\subset B$.
  5. Let $X = (S,\tau)$ and $X' = (S,\tau')$ be two topological spaces with the same underlying set $S$. Let $\id\colon X'\to X$ be the identity.
  6. Find a set $X$ and two topologies $\tau_{1}$ and $\tau_{2}$ on $X$ such that $(X,\tau_{1})$ and $(X,\tau_{2})$ are homeomorphic but $\tau_{1}$ and $\tau_{2}$ are not comparable.

Exercise sheet 1

  1. Show that a space $X$ is Hausdorff if and only if the diagonal $\Delta\coloneqq\{\, (x,x)\mid x\in X \,\}$ is closed in $X\times X$. Conclude that any metric space is Hausdorff.
  2. Give a subbasis for the product topology on $X\times X$ where $X = \mathbb{R}$ is equipped with the cofinite topology. Conclude that $X$ is not Hausdorff.
  3. Let $A\subset X$ be a subset of a topological space.
    • Prove that $\overline{A} = X\setminus(X\setminus A)^{\circ}$.
    • Show that repeatedly taking closures and complements of $A$ gives at most 14 distinct sets (including $A$).
  4. Let $X$ be a finite set. If $\preceq$ is a preorder on $X$, define a topology $\tau_\preceq$ on $X$ consisting of the upwards closed sets with respect to $\preceq$.
    • Check that $\tau_\preceq$ is a topology.
    • For any topology $\tau$, there is a preorder $\preceq$ with $\tau = \tau_\preceq$.
    • $\preceq$ is a partial order if and only if $\tau_\preceq$ is $T_0$.
    • $\tau_\preceq$ is already discrete if it is $T_1$.

Exercise sheet 2

  1. Let $X = \mathbb{R}$ with the topology of countable complements.
    • Characterise the sequential closure of any subset $A\subseteq X$.
    • Conclude that $X$ is not Fréchet-Urysohn.
  2. Give an example of a Fréchet space $X$ together with a sequence $x_1,x_2,\dots$ that has more than one limit point.
  3. Show that the projection maps $\pi_1\colon X\times Y\to X$ and $\pi_2\colon X\times Y\to Y$ are open maps.
  4. Let $X = \mathbb{R}^\mathbb{N}$ be the product space of countably many copies of $\mathbb{R}$ with the standard bounded metric given by $$\overline{d}(x,y) = \min \{ \lvert x-y \rvert, 1 \}.$$ Show that the product topology on $X$ is metrisable.
  5. Show that for any two disjoint closed subsets $A,B\subseteq X$ there exists a continuous function $f\colon X\to [0,1]$ with $f(A) = {0}$ and $f(B) = {1}$ if and only if $X$ is normal.

Exercise sheet 3

  1. Fix a prime number $p\in\mathbb{Z}$. For any $x\in\mathbb{Q}^\times$, the p-adic absolute value of $x$ is given by $$ \lvert x\rvert_p = p^{-\mathrm{val}_p(x)} $$ where $\mathrm{val}_p(x) = \max\left\{\, n \mid p^n \text{ divides } x \,\right\}$. This absolute value induces the p-adic metric on $\mathbb{Q}$ by $$\mathrm{d}_p(x,y) = \lvert x-y\rvert_p.$$ Prove the following statements.
    • The p-adic metric satisfies $\mathrm{d}_p(x,y)\leq\max\{\, \mathrm{d}_p(x,z), \mathrm{d}_p(y,z) \,\}$.
    • If $y\in\mathrm{B}(x,r)$, then $\mathrm{B}(x,r) = \mathrm{B}(y,r)$.
    • Any open ball $\mathrm{B}(x,r)$ is closed. (Hint: You may use without proof that $\lvert x+y\rvert_p = \max\{\lvert x\rvert_p, \lvert y\rvert_p\}$ if $\lvert x\rvert_p\neq\lvert y\rvert_p$.)
    • Any two open balls $\mathrm{B}(x,r)$ and $\mathrm{B}(y,s)$ are either disjoint or contained in one another.
    • The closed unit ball can be written as the union of open balls $$\overline{\mathrm{B}}(0,1) = \mathrm{B}(0,1)\cup\mathrm{B}(1,1)\cup\dots\cup\mathrm{B}(p-1,1).$$
    • The connected component of any point $x\in\mathbb{Q}$ is the singleton ${x}$.
  2. Let $X = \{\, n\in \mathbb{N} \mid x \geq 2 \,\}$ and consider the family of sets $$U_n = \left\{\, x\in X \mid n \text{ divides } x\,\right\}.$$
    • Show that the relation `$x$ divides $n$’ gives a preorder $n \preceq x$ on $X$, thus making $X$ into a topological space with the corresponding Alexandrov topology $\tau_\preceq$.
    • Show that $X$ is Kolmogorov but not Fréchet.
    • Characterise the isolated points of $X$.
  3. Find an equivalence relation $\sim$ on the unit square $[0,1]^2\subset\mathbb{R}^2$ with the subspace topology such that the torus $$T^2\coloneqq S^1\times S^1$$ is homeomorphic to the quotient space $X = [0,1]^2/\sim$.
  4. Construct a sequentially continuous map $f\colon\mathbb{R}^\mathbb{R}\to X$ for some topological space $X$ such that $f$ is not continuous. (Hint: Consider the subspace $2^\mathbb{R}$ instead. In there, construct the function $f$ by using the set of elements that assume a specific value only countably often.)

Exercise sheet 4

  1. Show that a sequence $a\colon\mathbb{N}\to X$ converges if and only if there is a continuous extension $\hat{a}$ to the one-point compactification $\hat\mathbb{N} = \mathbb{N}\cup{\infty}$, \ie there exists a continuous map $\hat{a}\colon\hat\mathbb{N}\to X$ such that $\hat{a}\vert_{\mathbb{N}} = a$.
  2. Let $X$ be first-countable. Show that $X$ is Hausdorff if every convergent sequence has a unique limit.
  3. Let $X$ be a topological space with the property that every compact set is closed. We say that $X$ is a KC-space. Prove the following statements.
    • $X$ is Frèchet.
    • Every convergent sequence in $X$ has a unique limit.
    • The one-point compactification $\hat\mathbb{Q}$ of the rationals (with the Euclidean topology) is a KC-space which is not Hausdorff.
  4. Let $X$ be the quotient space obtained from $[-1,1]$ by the identification $x\sim-x$ where $x\in(-1,1)$. Prove the following statements.
    • $X$ is homeomorphic to the the interval $[0,1]\cup{1’}$ with doubled right-endpoint, where a local basis of $1’$ is given by the sets $(a,1)\cup{1’}$.
    • $X$ is Fréchet, but not Hausdorff.
    • $X$ is compact. (Hint: Write $X$ as finite union of compact subspaces.)
    • There exist compact subsets of $X$ whose intersection is not compact.

Exercise sheet 5

to be determined