Re: Lectures WS 2024/25 - General Topology (2-MAT-211)
Posted: Tue Nov 26, 2024 2:51 pm
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
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