6TOPO Topology, winter semester 2022/2023

 HOMEWORKS

Homework 1
Give an example of topology on a 4-element set different from the trivial topology, from the discrete topology, and from the example in [V], Exercise 2.B.2.3(1) we did in the class.

Homework 2
Prove that a closed interval \([a,b]\) is closed in the standard topology on \(\mathbb R\).

Homework 3 ([V], Riddle 3.4)
What topological spaces have exactly one base?

Homework 4 ([V], 3.3)
Prove that any base of the standard topology on \(\mathbb R\) can be decreased.

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
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 7 ([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 8 ([V], 4.23)
Prove that in any metric space, any sphere is closed in the induced topology.

Homework 9 ([V], 4.21,4.22)
1) Give an example of a metric space and a closed ball in it which is open in the induced toplogy.
2) Give an example of a metric space and an open ball in it which is closed in the induced toplogy.

Homework 10
1) ([V], 4.K) Prove that a finite topology is metrizable iff it is discrete.
2) Provide an example of a metric on an infinite set, different from the 0-1 metric, such that the induced topology is discrete.
3) ([V], 4.J) For which cardinalities \(\kappa\) the trivial topology of cardinality \(\kappa\) is metrizable?

Homework 11 ([V], 4.32)
Is it true that if \(\rho_1\) and \(\rho_2\) are two metrics on \(X\), then \(min(\rho_1,\rho_2)\) and \(\rho_1\rho_2\) are metrics on \(X\)?

Homework 12
What can be said about a set \(X\) such that any two topologies on \(X\) are comparable (i.e., for any two topologies on \(X\), one of them is finer than another one?).

Homework 13
Consider the topology on a set \(X\) such that the closed sets are exactly finite subsets of \(X\). Prove that this is indeed a topology. Is this topology metrizable?

Homework 14
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 15
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 16 ([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 17 ([V], 10.7)
In the class we proved that a map \(f: X \to Y\) is continuous iff for any subset \(B \subseteq Y\), it holds \(Cl(f^{-1}(B)) \subseteq f^{-1}(Cl(B))\). Formulate and prove a similar criteria of continuity in terms of \(Int\) instead of \(Cl\).

Homework 18 ([V], 10.13)
Finish what we have started in the class: prove that continuous maps of the arrow into itself are exactly maps monotonically increasing and continuous from the left (in the sense of the standard topology, i.e., for example, \(\sup_{x \lt a} f(x) = f(a)\) at each point \(a\)).

Homework 19 ([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 20 ([V], 20.8)
Prove that if \(A\) is closed in \(X\), and \(B\) is closed in \(Y\), then \(A \times B\) is closed in \(X \times Y\).

Homework 21
What one can say about topological spaces \(X\) and \(Y\), if the topology on \(X \times Y\) is: a) discrete b) trivial?

Last modified: Wed Sep 13 17:49:33 CEST 2023