6/7TOPO Topology, winter semester 2024/2025

 HOMEWORKS

Rules for submitting homeworks
Scores

Homework 1
Prove that a half-open interval \([a,b)\) is neither open, nor closed in the standard topology on \(\mathbb R\).

Homework 2
Give an example of topology on a set \(X\) different from the discrete topology, such that there is a clopen set \(A\) different from the empty set and from \(X\).

Homework 3
Give example of a topological space having exactly:
1) 1
2) 2
3) 3
4) 4
bases.

Homework 4
A subset of a set \(X\) is called cofinite if its complement within \(X\) is finite. Prove that \(\{\text{the set of all cofinite subsets of } X\} \cup \{\varnothing\}\) is a topology on \(X\). Prove that any base of this topology can be decreased. Compare with the similar statement for the standard topology on \(\mathbb R\) we proved in the class, and try to formulate a common theorem generalizing these two cases.

Homework 5 ([V], 3.9)
Prove that all infinite arithmetic progressions consisting of positive integers form a base for some topology on \(\mathbb N\).

Homework 6
Consider the following topologies on \(\mathbb R\):
1) the standard topology;
2) the cofinite topology (i.e., closed sets are finite subsets and \(\mathbb R\), as in Homework 4);
3) the partition topology for some fixed partition of \(\mathbb R\).
How these topologies are related: which one is coarser and which one is finer?

Homework 7
Is the following function a metric on \(\mathbb R\)? \[ \rho(x,y) = \begin{cases} 0, & \text{ if } x = y \\ 1, & \text{ if } 0<|x-y| \le 1 \\ 2, & \text{ if } |x-y| > 1 . \end{cases} \]

Homework 8 ([V], 4.9)
Find a metric space and two balls in it such that the ball with the smaller radius contains the ball with the bigger one and does not coincide with it.

Homework 9 ([V], 4.23)
Prove that in any metric space, any sphere is closed in the induced topology.

Homework 10
Prove that the cofinite topology on an infinite set is not metrizable.

Homeworks due Tuesday November 5

Homework 11
Describe all situations when:
a) a subspace topology of a discrete topology is trivial;
b) a subspace topology of a trivial topology is discrete.

Homework 12
Let \(X\) be an arbitrary set, and \(Y \subseteq X\) its arbitrary subset. It is true that:
a) any topology on \(Y\) is a subspace topology of a topology on \(X\)?
b) any topology on \(Y\) is a subspace topology of the unique topology on \(X\)?

Homework 13
Describe the subspace topology on \(\mathbb Q\) induced from the cofinite topology on \(\mathbb R\).

Homework 14 ([V], 6.15)
Is it true that for any two sets \(A\) and \(B\):
1) \(Cl (A \cap B) = Cl(A) \cap Cl(B)\);
2) \(Cl (A \cup B) = Cl(A) \cup Cl(B)\)?

Homework 15 (question posed by Trey in the class)
Is it true that any open map is continuous?

Homework 16 ([V], 10.11)
Describe continuous maps from \(\mathbb R\) with cofinite topology to \(\mathbb R\) with the standard topology.

Homework 17
In the class we proved that a map \(f: X \to Y\) is continuous iff \(f^{-1}(Cl(B)) \supseteq Cl(f^{-1}(B)) \text{ for any } B \subseteq Y\). Prove equivalence of continuity with one of the remaining three conditions formulated in the class of your choice.

Homework 18 ([V], 10.13)
Describe continuous maps from the arrow to itself.

Homework 19 ([V], part of 10.18)
Let \(f,g: X \to \mathbb R\) be two continuous maps (\(X\) is an arbitrary topological space, and the topology on \(\mathbb R\) is standard). Prove that the map \(x \mapsto max\{f(x),g(x)\}\) is continuous.

Homework 20
Find the homeomorphism group of:
1) ([V], 11.6) the topological space on the 4-element set \(\{a,b,c,d\}\) given by the topology \(\{\{a,b,c,d\}, \varnothing, \{a\}, \{b\}, \{a,c\}, \{a,b,c\}, \{a,b\}\}\);
2) a topological space with partition topology of your choice (but different from the trivial or discrete topologies).

Homework 21 ([V], 11.35)
Prove that the spaces \(\mathbb Z\), \(\mathbb Q\) (with topology induced from the standard topology on \(\mathbb R\)), \(\mathbb R\) with the standard topology, \(\mathbb R\) with cofinite topology, and the arrow are pairwise non-homeomorphic.

Homework 22 ([V], 12.24)
Prove that the Cantor set is totally disconnected.

Homework 23 ([V], 12.31)
Let \(X\) be a topological group (i.e., a set having the structure both of a topological space and of a group, and such that the left and the right multiplication by any element of the group are continuous maps \(X \to X\)). Prove that the connected component of unity is a normal subgroup.

Homework 24 ([V], 13.3 modified)
Give a topological classification of the letters of the Czech alphabet: A, Á, B, C, Č, D, Ď, E, É, Ě, ..., Z, Ž regarded as subsets of the plane (the arcs comprising the letters are assumed to have zero thickness). Take into account all its pecularities (diacritics, letter Ch).

Homework 25 ([V], 13.9x)
Let \(G\) be a group equipped with a topology such that for each \(g \in G\) the map \(G \to G: x \mapsto xgx^{-1}\) is continuous, and let \(G\) with this topology be connected. Prove that if the topology induced on a normal subgroup \(H\) of \(G\) is discrete, then \(H\) is contained in the center of \(G\).

Homework 26
Does the Heine-Borel theorem (i.e., a subset is compact iff it is closed and bounded) holds in \(\mathbb R^n\) equipped with the metric \(\frac{\rho}{1 + \rho}\), where \(\rho\) is the usual Euclidean metric?

Homework 27
Let \((X, \Omega)\) be a topological space, \(X^* = X \cup \{c\}\) an one-element extension of \(X\), and \(\Omega^* = \Omega \cup \{(X \backslash C) \cup \{c\} \>|\> C \text{ is a closed and compact subset of } X\}\).
1) Prove that \((X^*, \Omega^*)\) is a compact topological space.
2) Prove that \([0,1)^* \simeq [0,1]\) (both topological spaces are considered with subspace topology from the standard topology).

Last modified: Tue Dec 17 2024 17:07:00 CET