Topology Exercise List

## Point-Set Topology

1. Let $$X$$ be a compact space and $$C\subseteq X$$ a closed subset. Prove that $$C$$ is compact.
2. Let $$X$$ be a topological space. Prove that all singletons $$\{x\}\subseteq X$$ are closed if and only if $$X$$ is $$T_1$$. Recall that $$X$$ is $$T_1$$ if for any distinct $$x\neq y$$ in $$X$$, there exist open sets $$U_x$$ and $$U_y$$ such that $$x\in U_x$$, $$y\in U_y$$, $$y\not\in U_x$$, and $$x\not\in U_y$$.
3. Suppose $$\{A,B\}$$ form a separation of a topological space $$X$$. If $$Y$$ is a connected subset of $$X$$, prove that either $$Y\subseteq A$$ or $$X\subseteq B$$.
4. Let $$X$$ be a connected space and $$f\colon X\to Y$$ be a continuous map. Prove that $$f(X)$$ is connected.
5. Prove that if $$X$$ is connected and locally path connected, then $$X$$ is path connected.
6. Let $$(X,d)$$ be a metric space. Prove that the metric $$d\colon X\times X \to X$$ is continuous with respect to the product topology on $$X\times X$$ of the metric topology on $$X$$.
7. Let $$X$$ be a topological space. Prove that $$X$$ is Hausdorff if and only if $$\Delta = \{(x,x)\} \subseteq X\times X$$ is closed.
8. Let $$p\colon E\to B$$ be a covering map where $$E$$ and $$B$$ are connected. Prove the following statements:
1. If $$E$$ is compact, then $$B$$ is compact.
2. If $$|p^{-1}(b_0)|=k$$ for some $$b_0\in B$$, then $$|p^{-1}(b)|=k$$ for all $$b\in B$$.
3. If $$B$$ is compact and $$p^{-1}(b)$$ is finite for some $$b\in B$$, then $$E$$ is compact.
4. If $$E$$ is compact, then $$p^{-1}(b)$$ is finite for all $$b\in B$$.

## Algebraic Topology

1. Let $$X$$ and $$Y$$ be path connected and semi-locally simply connected.
1. Prove that $$\pi_1(X\times Y) \cong \pi_1(X) \oplus \pi_1(Y)$$.
2. Prove that $$\pi_1(X\wedge Y) \cong \pi_1(X) * \pi_1(Y)$$.
2. If $$K$$ is a simplicial complex and $$p\colon L \to K$$ is a $$n$$-sheeted cover of $$K$$ by a simplicial complex $$L$$, then $$|\Delta_k(L)| = n\cdot |\Delta_k(K)|$$. In particular, $$\chi(L)= n\cdot \chi(K)$$.
3. Let $$X$$ be a connected graph and $$x\in X$$.
1. If $$x$$ is not a vertex, then $$H_1(X,X\backslash\{x\}) \cong \mathbb{Z}$$
2. If $$x$$ is a vertex, then $$H_1(X,X\backslash\{x\}) \cong \mathbb{Z}^n$$ where $$n$$ is the degree of the vertex $$x$$.
4. Let $$G_1$$ and $$G_2$$ be connected graphs. $$G_1$$ and $$G_2$$ are homeomorphic as topological spaces if and only if the number of valence $$k$$ vertices in $$G_1$$ is equal to the number of valence $$k$$ vertices in $$G_2$$ for all $$k\geq 3$$.
5. Let $$p\colon E\to B$$ be a finitely sheeted path connected cover of $$B$$. Prove that $|p^{-1}(b)| = [ \pi_1(B) : p_* \pi_1(E) ]$
6. Let $$G$$ and $$H$$ be any non trivial groups. Prove that the free product $$G*H$$ is not abelian.

## Differential Topology

1. Let $$M$$ be a manifold. Prove that for any $$p\in M$$, there are charts
1. $$(U_0,\phi_0)$$ such that $$\phi_0(U_0) = B_r( \phi_0(p) )\subseteq \mathbb{R}^n$$ for some $$r>0$$.
2. $$(U_1, \phi_1)$$ such that $$\phi_1(p)=0$$ and $$\phi_1(U_1)=B_1(0)$$.
3. $$(U_2, \phi_2)$$ such that $$\phi_2(p)=0$$ and $$\phi_2(U_2)=\mathbb{R}^n$$.
2. Prove that every manifold is locally path connected. Further, deduce that any connected manifold is path connected.
3. Prove that every manifold admits a basis of charts.
4. Prove that any compact $$n$$ manifold is not diffeomorphic to a subset of $$\mathbb{R}^n$$.
5. Let $$M$$ and $$N$$ be smooth manifolds with $$M$$ connected and $$f\colon M \to N$$ a smooth map such that $dF_p\colon T_p M \to T_{F(p)} N$ is trivial for all $$p\in M$$. Prove that $$F$$ is a constant map.