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.