Sendai Logic Mathematical Institute, Tohoku University, Sendai, Miyagi, Japan

#### 2020.11.06 Naruaki Kato and Tadayuki Honda

Weak König’s Lemma and Invariance of Domain in Second Order Arithmetic / Heine Borel compactness and Lindelof property in higher order arithmetic

• Date/Time: November 6, 2020 (Friday) / 15:00 - 17:00.

• Venue: Room 802, Science Complex A, Tohoku University.

• Speaker: Naruaki Kato (東北大学大学院 理学研究科)

• Title: Weak König’s Lemma and Invariance of Domain in Second Order Arithmetic

• Abstract: It is a well-known theorem that Weak König Lemma and Brouwer’s Fixed Point Theorem are mutually equivalent over the system $$\mathrm{RCA}_0$$ of second order arithmetic, which was proved by Shioji and Tanaka in 1990. In the proof, they constructed a retraction map from the square $$I^2$$ to its boundary under the assumption that Weak König Lemma fails. At this moment, the theorem Invariance of Domain also fails. In 2020, under the same assumption, Kihara proved that the invariance theorem does not hold not only squares but also higher dimension. In particular, he constructed a topological embedding of $$\mathbb{R}^4$$ into $$\mathbb{R}^3$$. However, he left the following open question: Does $$\mathrm{RCA}_0$$ prove that there exists no topological embedding of $$\mathbb{R}^3$$ into $$\mathbb{R}^2$$.
• Abstract: We already know that Heine Borel compactness for $$[0,1]$$ is equivalent to $$\mathrm{WKL}_0$$ over $$\mathrm{RCA}_0$$. But this “compactness” has a restricted meaning, i.e., coutable compactness. We talk about compactness for uncoutable covers and the relation between compactness and the Lindelöf property, based on the result of Dag Normann and Sam Sanders.