Topological Spaces starter
Topology introduction revisited
Description
If we have a set of points $X$, how can we make a precise notion of closeness and locality? We can define a notion of distance between individual points and have those notions follow as consequences. However, we can be more subtle and define whats known as a \emph{topology} on this set making $X$ \emph{topological space}, which makes precise those notions of closeness, locality, and therefore the notion of continuity (the preserving of closeness) in $X$ directly. Subsequent notions which can also be represented in this setting are that of connectedness (and therefore disconnectedness), compactness and limits.
Look at the beginnings of topology and topological spaces. We cover much of Munkres Chapter 2 and its exercises but with reflection and introspection. The ideas are known by all mathematicians and yet the presentation is considered too new for most university students but at the same time looking back on it now is quite strikingly out of date. The basics are still the same but they appear different, the focus is on the concrete spaces and less on the functions between them. Some perspective is added with category theory in mind but much of it is looking closely at the foundations with a classical perspective.
Lots of the earlier basic examples of topological spaces are examined in detail.
Product spaces, quotient spaces, subspaces are all defined and examined topologically.
Continuous functions, closed sets, open sets, Hausdorf space, T1 space, limit point, basis, base, sub base,
Metric spaces and metric topology is currently omitted.
Connectedness and compactness is omitted.
This is for beginners in topology but not necessarily beginners in mathematics especially if you have not used you mind much before.
What You Will Learn!
- Define a topological space, topology, open set, closed set, continuous function
- Understand and use the notion of a base or basis of neighbourhoods
- Various universal constructions including product, quotient and subspace topology
- Definition chase, basic proofs, solve simple exercises with unfamiliar terms
Who Should Attend!
- Students who are trying to grasp abstractions in maths at a high level
- Students who want to fill in gaps from their knowledge
- Students who want to be mathematicians
- Smart students