Editing Geometry (section)

==Topological spaces==
[https://www.youtube.com/watch?v=1wyOoLUjUeI&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=4 Lecture 04]

[https://www.youtube.com/watch?v=hiD6Tz06k30&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=5 Lecture 05]

topologies and topological spaces; examples; chaotic and discrete topologies; coarser (or weaker) and ﬁner (or stronger) topologies; open subsets; open balls; standard topology on R^d with proof; induced (or subset) topology with proof; product topology; sequences, converge and limit points; open neighbourhoods; definitely constant sequences; continuity of maps between topological spaces; examples; homeomorphisms and homeomorphic spaces.

Separations properties: T1, T2 (Hausdorff), T2 an a half; covers and open covers, subcovers and finite subcovers; compact spaces; Heine-Borel theorem (compact if and only if closed and bounded); open and locally finite refinements; paracompactness; metrisable spaces and Stone's theorem; long line (or Alexandroff line); partition of unity subordinate to an open cover; examples; connectedness and proof that M is connected if and only if M and the empty set are the only subsets which are both open and closed; path-connectedness and proof that path-connectedness implies connectedness; homotopic curves on a topological space; concatenation of curves; fundamental group; group isomorphism; topological invariants and classification of topological spaces; examples: 2-sphere, cylinder, 2-torus.