HOMEWORKS
Homework 2 (1 point).
Give an example of topology on a set \(X\) different from the trivial and the
discrete topology, having a set different from the empty set and from \(X\)
which is both closed and open.
Homework 8 (0.5 points) ([V], 5.2(3)).
Describe the topological structure induced on the 2-element set \(\{1,2\}\)
by the topology \(\mathbb R_{T_1}\) (recall that \(\mathbb R_{T_1}\) is the
topology on \(\mathbb R\) whose open sets are the empty set and complements to
all finite subsets of \(\mathbb R\)).
Homework 12 (1 point).
Give an example of a topological space \(X\) with a non-discrete topology, and
equivalence relation \(S\) on it, such that the quotient \(X / S \) is a
discrete topological space.
Homework 13 (1 point).
Let \(\ell_1, \ell_2\) be two lines in \(\mathbb R^2\). How many homotopically
nonequivalent paths are there in the space
\(\mathbb R^2 \backslash (\ell_1 \cup \ell_2)\)?
Created: Fri Nov 13 2020
Last modified: Mon Dec 5 17:31:53 CET 2022