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

[Martin Sleziak]

In this topic I will try to summarize what we talked about during the lectures. (It might be useful, for example, if you miss some of the lectures.)
I'd like to keep this particular topic only for this specific purpose - so if you have some other questions, it is better to star a new topic (or ask personally, by email, etc.)

You can have a look what we did in the previous years:
viewtopic.php?t=1884
viewtopic.php?t=1712

Videos (in Slovak) are available for some parts: viewtopic.php?t=1588

[Martin Sleziak]

1st week:

Lecture 1: (24.9.)
Introduction: A short introduction discussing at least very roughly what we would like to cover during this semester. It can be briefly summarized as:
* We will discuss several notions you already know in the context of metric spaces to a more general situation - for topological spaces. (The key topics include continuity, convergence and compactness.)
* I have shown a brief sketch of the proof based on nets that there is an element belonging to $\ell_\infty^*\setminus\ell_1$ - we will later (after having sufficient apparatus) prove this fact in detail. Another viewpoint: We have shown existence of something similar to a limit of a sequence - but in this case we assign a number to each bounded sequence, not only to the convergent sequences. (And I have mentioned that this illustrates a more general phenomenon; in many situation compactness can help us to show existence of some object based on suitable approximations of this object.)

Here is a link with various ways how to prove this fact: Dual of $l^\infty$ is not $l^1$. (E.g., based on Hahn-Banach Theorem. But in this lecture we are mainly interested in the proof Na tomto predmete nás ale viac bude zaujímať dôkaz, ktorý využíva kompaktnosť a konvergenciu sietí resp. filtrov.)

Topological spaces. Definition of a topological space.
We have briefly mentioned that on this subject we use the convention that intersection of the empty system is undefined. (But some texs use a different convention.)

Spoiler:

This issue is briefly described also in the text. And you can have a look here:

• Nullary intersection
• Empty intersection and empty union
• Why axiom 3) of topology is redundant?
• Why is this definition of topological space equivalent to the common one?

A few simple examples: Discrete and indiscrete subspace. Sierpiński space.
We described one way how to obtain from a metric space $(X,d)$ the corresponding topology $\mathcal T_d$. (We will talk about this once again after introducing the notion of basis for a topology.)

Lecture 2: (25.9.)
Subspace of a topological space. (We'll discuss this notion in more detail later.)
Closed sets and clopen sets. We described how a topology can be generated using closed sets.
Cofinite and cocountable topology.
Base for a topology. Definition. Characterization of a base, generating a topology using a base.
Topology determined by a metric, Sorgenfreyova line, Moore plane (In the text on the course website, Moore plane is defined using neighborhood basis. During the lecture I defined this topological space by describing a base. However, these two approaches are rather similar.)

[Martin Sleziak]

2nd week

Lecture 3: (1.10.)
Subbase. Definition and characterization of a subbase.
Neighborhood base. Definition of a neighborhood and of a neighborhood base. We have mentioned that a system of open sets $\mathcal B$ is a base iff it determines a neighborhood basis at each point - I skipped the proof. (The text on the website contains also a characterization of a neighborhood basis consisting of open sets - I have not discussed this part; we will often be using base when defining a topology - we already know that it suffices to verify the conditions $(B1)$ and $(B2)$; having a similar result of neighborhood bases would not simplify things too much.)
Closure. We defined the closure of a set, some basic properties and an equivalent characterization.
We mentioned without proof that the four conditions (CL1)-(CL4) are sufficient to define a topology using a closure operator.
We proved the result about the union of closures (and the union of closed sets) for locally finite systems.

Lecture 4: (2.10.)
Dense set. Definition of a dense subset, equivalent characterization (intersects every non-empty open set). For a dense set $D$ and an open set $U$ we have $\overline{U\cap D}=\overline U$.
Separable spaces, first and second countable spaces. We defined the notions: First countable space, secound countable space, separable space.
Every second countable space is first countable.
Every second countable space is separable. For metrizable spaces the converse implication is also true. (I.e., a separable metrizable space is second countable.)
Some examples and counterexamples: Sorgenfrey line and Moore plane are separable. Neither of these two spaces is second countable. (We will give the proof for Moore plane later.)
You can try to think about some know examples of Banach spaces: viewtopic.php?t=1583
We will later show that the quotient space $\mathbb R/\mathbb Z$ is another example of a separable space that is not second countable.
In the context of looking for counterexamples for various claims about topological spaces, I have mentioned the pi-base website: viewtopic.php?t=1718
Interior. We defined interior of a set, mentioned the relation to the closure and some basic properties. (We skipped the proofs.)

[Martin Sleziak]

3rd week:
Continuity. Continuity at a point, global continuity and characterization using preimages of open sets. Characterization of continuity using preimages of closed sets. Characterization of continuity using closure.
A continuous function maps dense set to a dense set. Continuous image of a separable space is separable.
Homeomorphisms.
Definition of a homeomorphism, basic properties.
Some examples of homeomorphic spaces - mostly subspaces of $\mathbb R$ and $\mathbb R^2$
Examples of homeomorphisms:
Homeomorphisms between intervals: $(0,1)\cong\mathbb R$, $(0,1)\ncong\langle0,1\rangle$.
Circle with one omitted point is homeomorphic to the real line.
We have mentioned stereographic projection, which yields a corresponding result for the n-dimensional sphere: že $S^n\setminus\{*\}\cong\mathbb R^n$. (But we did not do a detailed proof that this is indeed a homeomorphism.)
Closed and open maps. Definition and relation to homeomorphisms.
We have mentioned what it means that one topology is finer (coarser) than another topology.
We have shown that an intersection of a system of topologies is a topology.
I have very briefly mentioned what we plan to do next - a very rough idea how the initial topology and the final topology will be defined. (We will do this properly in the next lecture.)

[Martin Sleziak]

4th week:
Lecture 7: (15.10.)
Initial and final topology.$\newcommand{\inv}[1]{#1^{-1}}\newcommand{\Invobr}[2]{\inv{#1}[#2]}$
We defined the initial topology. We have shown that it can be described using the subbase $$\mathcal S=\{\Invobr {f_i}{U}; i\in I, U\in\mathcal T_i\}.$$ Characterization of the continuous maps into the space with the inital topology.
We defined the final topology. We have shown that the final topology can be desscribed as $$\mathcal T=\{U\subseteq X; (\forall i\in I)\Invobr{f_i}{U}\in\mathcal T_i\}.$$ Characterization of the continuous maps from the space with the final topology.
Quotient maps and quotient spaces.
Definition of a quotient map. (It is a special case of the final topology - now we have a single map.)
Thus we get characterization of continuous maps from the quotient space.
We briefly described how this corresponds to equivalence relations. (A surjective map gives an equivalence relation and vice-versa. Intuitively we can view this as "gluing" some points of the space together.)
Examples (circle, cylinder and torus, Möbius strip, Klein bottle.) Some pictures and animations can be found here: viewtopic.php?t=1897
Characterization of quotient maps using closed sets. Composition of two quotient maps is a quotient map.
The space $\mathbb R/\mathbb Z$ is separable but not first countable.

Lecture 8: (16.10.)
Product space.
We started with the product of two topological spaces - although we alter shown the same facts for product of arbitrary system.
Definition of Cartesian product of (infinitely many) sets, projections, notation for some maps.
Product was defined as the coarsest topology such that all projections are continuous. (I.e., it is the initial topology w.r.t. $\{p_i; i\in I\}$.)
This gives the description of the subbase and characterization of continuous maps into the product. (And as a consequence we get the results about the continuity of $\langle f_i\rangle$ and $\prod f_i$.)
Projection is an open continuous map. (But it is not necessarily closed.)
Box topology - if we took the basis with prescribed open sets for all indices (not only finitely many) we would get a different topology, it does not have such nice properties.
We have mentioned that product of countably many first (second) countable space is first (second) countable.
We started the proof that product of $\mathfrak c$ separable space (Hewitt-Marczewski-Pondiczery theorem). So far we have shown this claim for product of $\mathfrak c$-many countable discrete spaces - this should cover the most relevant part of the proof.

[Martin Sleziak]

5th week:
Lecture 9: (15.10.)
Product space. We finished the proof that product of $\mathfrak c$-many separable space is separable.
We summarized once again the reasons why the product topology defined in this way is more natural than the box product.
Sum of topological spaces. We talked briefly about topological sum. We only explained the definition and mentioned some basic facts without going into details. (This construction is much simpler than the product - I have mentioned it mainly to show that these two constructions are, in some sense, dual to each other.)
Limit of a sequence. Definition of a limit of a sequence and some very simple examples. (And also a few remarks related to the fact that in general we do not have uniqueness.)
We proved that in Hausdorff spaces a sequences has at most one limit. (So we do not have problems with uniqueness in such spaces.)
Relation of convergence to closure and closedness, continuity and sequential continuity. (In both cases we saw that one implications is valid in arbitrary topological space. We proved that the opposite implication is true in first countable spaces - as a special case we get that it holds for metric spaces.)
The space $C(\omega)$.
We defined the space $C(\omega)$. We saw that it is homeomorphic to $\{0\}\cup\{\frac1n; n\in\mathbb N\setminus\{0\}\}$ (with Euclidean metric, i.e., as a subspace of the real line). We showed that convergence of a sequence $x\colon\mathbb N\to X$ can be characterized as continuity of the corresponding map $\overline x\colon C(\omega)\to X$.

Lecture 10: (23.10.)
Nets.
Definition of a directed set. Definition of a net and a limit of a net.
Limit of a net and subbase.
The net $(x_U)_{U\in\mathcal B_a}$ with $x_U\in U$ converges to $a$.
Characterization of closure and characterization of closed sets using nets. (Limits of nets completely determine the topology.)
Uniqueness of limit - characterization of Hausdorff spaces using nets.
Characterization of continuity using nets.
Counterexamples.$\newcommand{\R}{\mathbb R}$
We have seen an example showing that sequences are, in general, not sufficient to describe closure of a set in topological spaces. (I.e., the characterization of the closure using nets is no longer true if we replace nets with sequences.)
We have used as a counterexample the space $\{0,1\}^{\R}$ (Cantor cube).
For the set $A=\{\chi_F; F\text{ is a finite subset of }\R\}$we have that $\chi_{\R}\in\overline A$. But no sequence of elements of $A$ converges to $\chi_{\R}$.
Another example is $\omega_1+1=\langle0,\omega_1\rangle$ with the order topology. (We did not show this example during the lecture - if you have good knowledge of ordinals you might have a look at this example, too.)
In this space, the subset $A=\langle0,\omega_1)$ is closed w.r.t. limits of sequences. But this set is not close; the net $x_\alpha=\alpha$ converges to $\omega_1$. (The symbol $\omega_1$ denotes the first uncountable ordinal.)
If we use the space $C(D)$ where we take $(\omega_1,\le)$ as the directed set $D$; the counterexample works practically the same as in the order topology.

[Martin Sleziak]

6th week:
Lecture 11: (29.10.)
Convergence of nets.
Nets and the space $C(D)$. We saw that convergence of a net $x\colon D\to X$ can be characterized as continuity of the corresponding map $\overline x\colon C(D)\to X$.
Convergence of nets in the initial topology. Convergence in the product space is the pointwise convergence.
Subnets.
Definition of a subnet, relation to the space $C(D)$.
Cluster points of a net - definition and characterization using subnets.
The set of cluster points is equal to $\bigcap\limits_{d\in D}\overline{\{x_e; e\in D, e\ge d\}}.$ (And consequently it is a closed set. We have skipped the proof of this fact - but this is basically just rewriting the definition.)
In literature you can find also different definitions of a subnet from the one we used here.
Riemann integral. We briefly mentioned that Riemann integral can be interpreted as a limit of a net. See also: What's the "limit" in the definition of Riemann integrals?
Uncountable sums. I have mentioned that the sum $\sum_{i\in I} x_i$ for an arbitrary index set can also be defined using nets. If you're interested in this notion, you can find some brief description in the text on the course website.
More details can be found in the text for the course Applications of set theory. Some further links and references are mentioned also here: viewtopic.php?t=1906

Lecture 12: (29.10.)
Filters, ultrafilters, F-limit. TODO

[Martin Sleziak]

7th week:$\newcommand{\FF}{\mathcal F}\newcommand{\Flim}{\operatorname{\FF-lim}}$
Lecture 13: (5.11.)
We have mentioned the space $C(\mathcal F)$ and the $\mathcal F$-limita of the function $f$ is related to the continuity of the function $\overline f\colon C(\mathcal F)\to X$. (We only mentioned this very briefly - the result and the proof are very similar as for the nets.)
Convergence of filters
Definition of the limit of a filter on $X$, relation the the $\mathcal F$-limit. (It is a special case of $\mathcal F$-limit for $f=id_X$. At the same time we have: $a\in \Flim f$ $\Leftrightarrow$ $f_*[\mathcal F]\to a$.)
Limit for a finer filter. Limit can be equivalently described using a subbase.
Characterization of closure, continuity, Hausdorff spaces using filters.
Initial topology (product topology) and filters. (In this case only sketched the proof.)
Cluster point of a filter - definition and basic properties. (It still remains to show, that cluster points can be characterized using finer filters and using ultrafilters.)

Lecture 14: (6.11.)
Cluster points of a filter. A point is a cluster point of $\FF$ if and only if it is a limit of a finer filter (finer ultrafilter).
T0, T1 a T2-spaces
Definitions of these classes of spaces. $T_1$-priestory are precisely the spaces where singletons are closed. They can be equivalently characterized as the spaces where constant nets (principal ultrafilters) have unique limit.
Subspace/product of $T_2$-spaces is again $T_2$. (In fact, we have shown a slightly more general version: We have proved this for the initial topology w.r.t. a family of mappings that separates points.)
A topological space is Hausdorff if and only id the diagonal $\Delta=\{(x,x); x\in X\}$ is closed in $X\times X$.
The set $\{x\in X; f(x)=g(x)\}$ is closed for continuous maps $f,g\colon X\to Y$, assuming $Y$ is Hausdorff. As a consequence we get that if $f|_D=g|_D$, for a dense set $D$, then $f=g$.
Regular spaces
Definition of a regular space and $T_3$-priestoru. Characterization via existence of a neighborhood such that $b\in U\subseteq\overline U\subseteq V$. (We haven't yet shown a corresponding result for a subbase.)

[Martin Sleziak]

8th week:
Lecture 15: (12.11.)
$\newcommand{\inv}[1]{#1^{-1}}\newcommand{\Invobr}[2]{\inv{#1}[#2]}\newcommand{\ol}[1]{\overline{#1}}$We have shown that for any continuous map we have $\ol{\Invobr fB}\subseteq \Invobr f{\ol{B}}$. (In fact, this is equivalent to continuity - but we only proved this implication. We did a proof using nets.)
Regular space
In the characterization with $b\in U\subseteq\overline U\subseteq V$ it suffices to take $V$ from a subbase.
Subspaces and products of regular spaces.
We mentioned that every metric space is regular. (We will prove a stronger fact that every metric space is completely regular.)
I did not include an example of a space which is $T_2$ but not $T_3$. (Such an example can be found in the lecture notes.)
Completely regular spaces$\newcommand{\inv}[1]{#1^{-1}}\newcommand{\Invobr}[2]{\inv{#1}[#2]}$
Definition of competely regular and Tychonoff space ($T_{3\frac12}$-priestoru).
In the definition we can replace $C(X,\mathbb R)$ by $C(X,I)$, i.e., use only the functions into the interval $I=\langle0,1\rangle$.
In connection with this we mentioned that for any operation which is continuous as a map $\mathbb R\times\mathbb R\to\mathbb R$ (such as addition or maximu) then using the composition with $\langle f,g\rangle$ we can show that from two continuous fuctions we get again a continuous function.
Characterization of completely regular spaces using open sets and using a subbase.
This class of spaces is closed under subspaces and products.
Every Tychonoff space is homeomorphic to a subspace of some Tychonoff cube, i.e., a space of the form $I^A$. (Specifically we can take $A=C(X,I)$ and we obtain the embedding using evaluation of functions.)

[Martin Sleziak]

9th week:
Lecture 16: (19.11.)
Completely regular spaces.
We proved a characterization of Tychonoff spaces as subspaces of Tychonoff cubes. Later we will show somewhat similar result: Compact $T_2$ spaces are precisely the closed subspaces of Tychonoff cubes.
Every metric space is a Tychonoff space.
Normal spaces.
Definition of a normal space and a $T_4$-space. Equivalent characterization using $C\subseteq O\subseteq\overline O\subseteq U$.
A closed subspace of a normal space is normal. (We have mentioned that the class of normal spaces is not closed under products and subspaces - at the moment without a proof.)
Urysohn's lema - with a proof. (Is a corollary we get that every $T_4$-space is Tychonoff.)
Every metrizable space is normal (and, consequently, $T_4$.)
The implications $T_4\Rightarrow T_{3\frac12} \Rightarrow T_3 \Rightarrow T_2$ cannot be reversed; so far we haven't shown any counterexamples. If the time permits, we might return to some of the examples later.
Similarly, we postponed the proof that normal spaces are closed neither under products nor under subspaces.

Lecture 17: (20.11.)
Normal spaces.
I have stated Tietze extension theorem without proof. We will probably get back to this result and show the proof later.
Compact spaces.
Definition of a compact space, simple examples. The interval $\langle0,1\rangle$ is compact.
In connection with this I have mentioned a proof method called real induction. (We will briefly mention it again when studying connected spaces.)
Characterization of compactness using finite intersection property. Closed subspace of a compact space is compact. Compact subspace of a $T_2$-space is closed. Every compact $T_2$-space is also a $T_4$-space.
Continuous image of a compact space is compact. (I will mention some corollaries of this result in the next lesson.)