
Explain how a topology generated by a basis equals all unions of basis elements, and is the smallest topology containing the basis, as connected through T1 and T3.
Explore five topologies on the real line - standard, upper and lower limit, polynomial, and others - by analyzing bases, open sets, and their comparisons, using lemmas 13.2 and 13.3.
Explore examples of topology, including discrete and order topologies, and learn how open rays form a subbasis and generate a basis for topologies.
Explore forming the product topology from bases: combine a basis for X with a basis for Y to generate open rectangles and study projection continuity.
Explore continuity in topological spaces by defining preimages of open sets as open, using basis and subbasis concepts, and relating to epsilon-delta and real analysis examples.
Explore how a limit point interacts with continuity, open sets, and preimages, and examine when a constant or injective function preserves limit points.
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is considered too new for most university students but at the same time looking back on it now is quite strikingly out of date.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.