HOMEWORKS
Homework 2.
Prove that a half-open interval \([a,b)\) is neither open, nor closed in the
standard topology on \(\mathbb R\).
Homework 6 ([V], 4.I).
Prove that a set U is open in a metric topology if and only if, together
with each of its points, the set U contains a ball centered at this
point.
Homework 7 ([V], 4.20, 4.23).
Prove that any closed ball and any sphere in a metric space are closed in the
induced topology.
Homework 8.
Give an example of a non-metrizable topological space different from those
considered in the class (i.e., finite topological spaces).
Homework 9 ([V], 4.32).
Is it true that if \(\rho_1\) and \(\rho_2\) are metrics on the same set \(X\),
then the function
\[
\rho(x,y) = \begin{cases}
\frac{\rho_1(x,y)}{\rho_2(x,y)}, &\text{ if } x \ne y \\
0 , &\text{ if } x = y
\end{cases}
\]
is a metric on \(X\)?
Homework 13.
Describe the topology on \(\mathbb Q\) induced from the cofinite topology on
\(\mathbb R\).
Homework 14 ([V], 6.16).
Fint the interior, closure, and boundary of \([0,1]\) and \((2,+\infty)\) in
the arrow.
Homework 15 ([V], 6.18).
Find the interior, closure, and boundary of \(\mathbb N\), \((0,1)\), and
\([0,1]\) in \(\mathbb R\) with respect to the cofinite topology.
Created: Sat Sep 25 2021
Last modified: Mon Nov 21 17:32:13 CET 2022