Lectures WS 2024/25 - General Topology (2-MAT-211)

[Martin Sleziak]

10th week:
Lecture 18: (26.11.)
Continuous image. Last week we have shown that continuous image of a compact space is compact. Now we looked at some corollaries of this fact. (If we assume that codomain is Hausdorff, we get a closed map. If it is, additionally, a bijection, we get a homeomorphism. In the case that the codomain is $\mathbb R$, the image is a closed and bounded. Consequently, a real-valued function on a compact space is bounded and it attains the maximum and the minimum.
Compactness and convergence.
In a compact space for any ultrafilter there is an $\mathcal U$-limit. Equivalent characterization of compactness using convergence of ultrafilters and existence of cluster points of filters.
A topological space is compact if and only if every net has a cluster point. (Equivalently: Every net has a convergent subnet.)
Tychonoff's theorem.
We mentioned two proofs Tychonoff's theorem. On proof was based on $\mathcal F$-limits.
Another proof is based on Alexander subbase theorem.. (But I skipped the proof of Alexander subbase theorem - I only showed how it can be used to prove Tychonoff's theorem. Proof of Alexander subbase theorem can be found in various places. I use this proof to illustrate application of Zorn's Lemma in the subject Applications of set theory; a proof can be also found in the notes for this subject.)

Lecture 19: (27.11.)
TODO

[Martin Sleziak]

11th week:
Lecture 20: (3.12.)
Connected spaces. Definition of a connected space and equivalent condition. The closed unit interval $\langle0,1\rangle$ is connected.
If the space $X$ contains a connected dens subset hten $X$ is connected. Continuous image of a connected space is connected.
If there are connected subsets such that $\bigcap\limits_{i\in I} S_i\ne\emptyset$ then the union $\bigcup\limits_{i\in I} S_i$ is a connected subset.
Characterization of connectedness using chains.
Product of connected spaces is connected. We have proved this for finite products - in the next lecture we will deal with arbitrary products.

Lecture 21: (4.12.)
Connected spaces. Product of connected spaces is connected.
Connected components. We defined connected components and showed that they form a decomposition of $X$. All components are closed.
Komponenty súvislosti sú uzavreté.
Path-connected spaces. Path-connected and arc-connected spaces.
Every arc-connected space is path-connected. We mentioned (without proof) that the reverse implication is true for $T_2$-spaces.
Every path-connected space is connected. Topologist's sine curve is an example of a space which is connecte but not path-connected.
Path-connected components. Product of path-connected space is path-connected. Continuous image of a path-connected space is path-connected.

[Martin Sleziak]

12th week:
Lecture 21: (10.12.)
Locally connected spaces.
Locally connected and locally path-connected spaces.
Broom space: Example of a space which is connected and linearly connected, but it is neither locally connected nor locally linearly connected.
A connected and locally path-connected space is path-connected.
A topological space is locally connected if and only if connected components of every open sets are open.
Quotient space of a locally connected space is locally connected.
Normal spaces
Jones' lemma. (Let $X$ be a normal space, $D$ be a dense subset of $X$ and $C$ be a closed subset of $X$ which is discrete (in the relative topology). Then $2^{|C|}\le 2^{|D|}$.)
Moore plane $\Gamma$ is a completely regular space which is not normal.
Other examples which we haven't done (yet): Jones' lemma can also be used to show that $\mathbb R_l\times\mathbb R_l$ is not normal. And based on this we can show that normal spaces are closed neither under subspaces nor under products.
(We will see whether we might get back to this - depending on the time and on the choice of other topics.)

Lecture 22: (11.12.)
Locally compact spaces.
We dealt only with the Hausdorff case. We mentioned several equivalent conditions for local compactness.
$\mathbb R$ is an example of a locally compact space. $\mathbb Q$ is not locally compact.
Open subspace and close subspace of a locally compact space is again locally compact. Without proof we mentioned that a subspace is locally compact iff it is an intersection of and open and closed subset.
One-point compactification.
We showed that every locally compact $T_2$-space has one-point compactification (Alexandroff compactification).
Examples: $C(\omega)$, circle as the one-point compactification of the real line. (Or sphere as the one-point compactification of the plane.|
Definition of a compactification. A space $X$ has a compactification iff and only if $X$ is Tychonoff.
We have mentioned that there exists also Stone-Čec compactification - but without mentioning any details. (I only mentioned that it is basically the compactification obtained in the proof about Tychonoff spaces.)

[Martin Sleziak]

13th week:
Lecture 23: (17.12.)
Paracompact spaces.
Refinement, definition of a paracompact space.
Every compact $T_2$-space is paracompact. Every discrete space is paracompact.
We proved four equivalent conditions to paracompactness.
Lecture 23: (17.12.)
Paracompact spaces.
Every paracompact $T_3$-space is normal.
Definition of partion of unity and locally finite partition of unity. A $T_3$-space is paracompact if and only if for every open cover there exists a subordinate partition of unity.
We have only mentioned (without a proof) that every metrizable space is paracompact.