Tychonoff theorem implies Axiom of choice

By | April 19, 2015
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이번 글에서는 Tychonoff theorem이 Axiom of choice를 함의한다는 것을 보이고자 한다. Tychonoff theorem은 다음과 같다.

Theorem. If \{X_i\}_{i\in I} is a family of compact spaces, then its product space \prod_{i\in I} X_i is compact space. 

이 명제의 Proof은 Axiom of choice에 의존한다. 이 글에서는 Tychonoff theorem을 증명하는 것이 목표가 아니다. 이 정리를 증명하는 방법은 다양한 형태의 Axiom of choice를 사용해서 증명하는 것이 알려져있다. 이 명제를 증명하는 방법으로는 Alexander subbase theorem을 사용하는 것이 쉽다.

이 글에서 보이고자 하는 사실은 다음과 같다.

Proposition. Axiom of choice is equivalent to the Tychonoff theorem.

Axiom of choice도 여러가지 variation이 많다. 이 글에서 Axiom of choice는 다음과 같다.

Axiom of choice. If \{X_i\}_{i\in I} is a family of nonempty sets, then \prod_{i\in I} X_i is nonempty.

Axiom of choice의 동치조건에 대해서 궁굼한 사람은 계승혁 교수의 강의록을 참고하라. 이제 명제를 증명한다.

Proof. \{X_i\}_{i\in I}를 nonempty set들로 이루어진 모임이라고 하자. 주어진 집합들에는 없는 원소 \omega \in \bigcup_{i\in I} X_i를 하나 잡자.  이제 Y_i=X_i \cup \{\omega}, \mathcal{T}_i = \{\varnothing, \{\omega\},Y_i\}라 놓으면 (Y_i, \mathcal{T}_i)는 topological space이고 열린집합이 유한개밖에 없으므로 compact space다. 그러므로 Tychonoff theorem에 의하여 \prod_{i\in I} Y_i는 compact이다.

\{\omega\}는 open set이므로 \pi^{-1}_{i}\left(\{ \omega\}\right)^c\prod_{i\in I} Y_i에서 닫힌 집합이다. 그리고 \{\omega\}^c = X_i이다. A_i = \prod^{-1} (X_i)라 놓자. Compact set의 성질을 이용하기 위하여 \{A_i\}_{i\in I}가 F.I.P를 갖는다는 것을 보이고자 한다.

유한 부분집합 J\subset I을 하나 잡자. 그러면 J=\varnothing인 경우에는 \bigcap_{i\in J} A_i = \mathcal{U}이므로 \omega \in \bigcap_{i\in J} A_i이다. J\neq \varnothing일 때는 각각의 j\in J에 대하여 x_j \in X_j를 하나 잡자. 그리고 다음과 같이

    \[ b_i = \begin{cases} x_i & i\in J \\                                         \omega & i\neq J \end{cases} \]

라 정의하자. 그러면 (b_i)_{i\in I} \in \bigcap_{j\in J} \pi^{-1}_{j} (X_j) =\bigcap_{j\in J} A_j이다. 그러므로 어떤 finite subset J\subset I를 잡아도 \bigcap_{j\in J} A_j \neq \varnothing이다. 그러므로 \{A_i\}는 F.I.P를 갖는다.

그런데 \prod Y_i가 compact set이므로 \varnothig\neq \bigcap A_i =\bigcap \pi^{-1} (X_i) = \prod X_i이므로 증명이 끝난다. \Box

One thought on “Tychonoff theorem implies Axiom of choice

  1. Duci

    12:00-1:30 Martin Bendersky The Conner-Flyod phenomenon The two lectures will be devoted to answering the plaintive question: What is the Conner-Floyd theorem (from 1966)? There are quite a number of proofs. I will give a (classic) proof that uses the Landweber exact functor theorem. Partly to remind the group what the theorem says. It is easier to state and prove the theorem in the context of BP. So in order to keep everything rather simple I will recall how BP is related to MU_* (the bordism of weakly almost complex manifolds). I will also recall some of the constructions in K-theory (e.g the K-theory Thom class and the Adams splitting of KU). The description of the CF theorem is rather algebraic. In particular the theorem gives a relation between complex cobordism and KU homology which is not as intuitive as KU cohomology. I will try to put together an argument that is more geometric, using a bordism description of KU homology. I hope I can restrict the only appearance of KU homology is the KU fundamental homology class of a U-manifold. 2:00-3:30 Martin Bendersky The Conner-Flyod phenomenon The two lectures will be devoted to answering the plaintive question: What is the Conner-Floyd theorem (from 1966)? There are quite a number of proofs. I will give a (classic) proof that uses the Landweber exact functor theorem. Partly to remind the group what the theorem says. It is easier to state and prove the theorem in the context of BP. So in order to keep everything rather simple I will recall how BP is related to MU_* (the bordism of weakly almost complex manifolds). I will also recall some of the constructions in K-theory (e.g the K-theory Thom class and the Adams splitting of KU). The description of the CF theorem is rather algebraic. In particular the theorem gives a relation between complex cobordism and KU homology which is not as intuitive as KU cohomology. I will try to put together an argument that is more geometric, using a bordism description of KU homology. I hope I can restrict the only appearance of KU homology is the KU fundamental homology class of a U-manifold.

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