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 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 topology.
2) Give an example of a metric space and an open ball in it which is closed in
the induced topology.
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 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 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: Tue Nov 19 2024 20:06:31 CET