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.

Homeworks due Tuesday November 19

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.

Homeworks due Tuesday November 26

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.

Last modified: Tue Nov 19 2024 20:33:07 CET