On successful completion of this module you should be able to:
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<ul>
<li>Establish the invariance of the Euler characteristic of a sphere, and compute simple Euler integrals.</li>
<li>Give the definitions of a topological space and a continuous map between topological spaces, and provide examples.</li> 
<li>Understand connectedness and compactness as topological invariants.</li>
<li>Understand the concept of homeomorphism, and use topological invariants to prove that certain spaces are not homeomorphic.</li>
<li>Construct new topological spaces using the subspace and quotient constructions.</li>
<li>Understand and represent simplicial complexes and triangulated spaces.</li>
<li>Understand homotopy equivalence, and informally explain why the Euler characteristic is a homotopy invariant of a triangulated space.</li>
<li>Prove the fundamental theorem of algebra, Perron’s theorem, Brouwer’s Fixed Point Theorem.</li> 
<li>Understand John Nash’s proof of the existence of Nash equilibria in game theory.</li>
<li>Understand the basic idea behind topological data analysis.</li>
</ul>

