Tychonoff's Theorem for Hausdorff Spaces
Theorem
Let:
- $I$ be an indexing set.
Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of non-empty Hausdorff spaces.
Let $\ds X = \prod_{i \mathop \in I} X_i$ be the corresponding product space.
Then $X$ is compact if and only if each $X_i$ is compact.
Proof
 | This article, or a section of it, needs explaining. In particular: Missing reference to the use of Boolean Prime Ideal Theorem/Ultrafilter Lemma or Axiom of Choice/Zorn's Lemma You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
First assume that $X$ is compact.
From Projection from Product Topology is Continuous, the projections:
- $\pr_i : X \to X_i$
are continuous.
From Continuous Image of Compact Space is Compact, it follows that the $X_i$ are compact.
Assume now that each $X_i$ is compact.
By Equivalence of Definitions of Compact Topological Space it is enough to show that every ultrafilter on $X$ converges.
Thus let $\FF$ be an ultrafilter on $X$.
From Image of Ultrafilter is Ultrafilter, for each $i \in I$, the image filter $\map {\pr_i} \FF$ is an ultrafilter on $X_i$.
Each $X_i$ is compact by assumption.
So by definition of compact, each $\map {\pr_i} \FF$ converges.
From Filter on Product of Hausdorff Spaces Converges iff Projections Converge, $\FF$ converges.
So, as $\FF$ was arbitrary, $X$ is compact.
$\blacksquare$
Boolean Prime Ideal Theorem
This theorem depends on the Boolean Prime Ideal Theorem (BPI), by way of Equivalence of Definitions of Compact Topological Space.
Although not as strong as the Axiom of Choice, the BPI is similarly independent of the Zermelo-Fraenkel axioms.
As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.
However, Equivalence of Definitions of Compact Topological Space only invokes the Ultrafilter Lemma, so this theorem is safe to use when proving the Boolean Prime Ideal Theorem from the Ultrafilter Lemma.
 | This page has been identified as a candidate for refactoring of advanced complexity. In particular: Messy and unsatisfactory way to organise this by hiding a crucial fact in the depths of a proof of arbitrarily arranged definitions of a basic mathematical definition, so we really need to apply some joined-up thinking to this Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code. |