### Questions (317)

 9 Prove that the min and max of 2 continuous function are continuous 8 closed and open set - set $S$ is open if and only if its complement is closed? 7 Limit superior inequalities proof: $\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge e$ 7 Find the error in following reason $(-z)^2=z^2 \implies \log(-z)^2=\log(z)^2 \implies2\log(-z)=2\log(z)\implies \log(-z)=\log(z)$ 6 Explain why $U_{44} \cong (\mathbb{Z}_{10} \oplus \mathbb{Z}_2)$.

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 +5 Find a map of the solid torus into itself having no fixed point. Where does the proof of the Brouwer theorem fail. +5 Find an example of a compact space which is not locally compact. -2 Find a necessary and sufficient condition for the Cartesian product $G \times H$ is Eulerian, for $G$ and $H$ are non trivial connected graphs. +5 Prove that $f_n$ converges uniformly on $[a,b]$

### Answers (5)

 3 Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation. 2 Prove that if $u$ and $v$ do not lie on a common cycle then $od(u)≠od(v)$ 1 Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a neighborhood of $f(Z)$( more detail) 1 Prove that if $G$ is an nontrivial connected graph with at most 2 bridges, then there exists an orientation $D$ 0 Prove or disprove: if $G$ is a graph such that $\chi(G) \leq \chi(S_k)$ then $G$ is embedded in $S_k$

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 4 differential-topology × 23 0 general-topology × 26 3 graph-theory × 110 0 abstract-algebra × 18 0 calculus × 90 0 real-analysis × 11 0 proof-writing × 45 0 group-theory × 9 0 complex-analysis × 29 0 sequences-and-series × 8

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