Chapter 5

(g) Let \(X\) be a topological space and let \(f: X \rightarrow Y\) be a surjective map. Let \(\mathscr{U}_{\mathrm{f}}\) denote the quotient topology on \(\mathrm{Y}\). Suppose that \(\mathscr{U}\) is a topology on \(\mathrm{Y}\) so that f: \(\mathrm{X} \rightarrow \mathrm{Y}\) is continuous with respect to this topology. Prove that if \(\mathrm{f}\) is a closed or an open mapping then \((\mathrm{Y}, \mathscr{U})\) is homeomorphic to \(\left(Y, \mathscr{U}_{f}\right)\). Furthermore. give examples to show that if \(\mathrm{f}\) is nejther open nor closed then \((\mathrm{Y}, \mathscr{H}) \neq\left(\mathrm{Y}, \|_{1}\right)\).

(b) The Mathe a model of a cylincler and a Mobius strip by using slips of paper. say \(40 \mathrm{~cm}\) by \(4 \mathrm{~cm}\). Draw a pencil line midway between the edges of the cylinder and of the Mobius strip. Now cut along the pencil lines. What is the result in each case? What if we cut along a line one-third of the distance between the edges?

(d) Let \(\mathrm{G}\) act on \(\mathrm{X}\) and define the stabilizer of \(\mathrm{x} \in \mathrm{X}\) to be the set $$ G_{x}=\\{g \in G ; g \cdot x=x\\} $$ Prove that \(\mathrm{G}_{\mathrm{x}}\) is a suógroup of \(\mathrm{G}\).

(e) Let \(\mathrm{G}\) act on \(\mathrm{X}\) and define the orbit of \(\mathrm{x} \in \mathrm{X}\) to be the subset $$ G \cdot x=\\{g \cdot x: g \in G\\} $$ of \(\mathrm{X}\). Prove that two orbits Gix, G.y are either disjoint or equal. Deduce that a Ci-set X decomposes inte a union of disjoint subsets.

(a) Let \(\mathrm{X}\) be the infinite strip \(\left\\{(\mathrm{x}, \mathrm{y}) \in \mathrm{R}^{2} ;-1 / 2<\mathrm{y}<1 / 2\right\\}\) in \(\mathrm{K}^{2}\) wilh \(Z\) acting on it by \(m \cdot(x, y)=\left(m+x,(-1)^{m} x\right)\). Show that the yuntient space \(X / Z\) is homeomorphic to the Mobius strip.

(c) Construct examples to show that if \(X\) and \(Y\) are topological spaces with \(\mathrm{G}\) acting on them such that \(\mathrm{X} / \mathrm{G} \simeq \mathrm{Y} / \mathrm{G}\) then \(\mathrm{X}\) and \(\mathrm{Y}\) are not necessarily homeomorphic.

(b) Let \(X\) be a G-space with \(G\) finite. Prove that the natural projection \(\pi: \mathrm{X} \rightarrow \mathrm{X} / \mathrm{G}\) is a closed mapping.