Much taken from [wikipedia](https://en.wikipedia.org/wiki/Martin's_axiom) # Statement For any cardinal $\kappa$ we define a statement, denoted by MA($\kappa$): For any [[forcing|partial order]] $P$ satisfying the [[Chain conditions|countable chain condition]] (hereafter ccc) and any family $D$ of dense sets in $P$ such that $|D|\leq \kappa$, there is a [[filter (mathematics)|filter]] $F$ on $P$ such that $F\cap d = \varnothing$ for every $d\in D$. >[!note] Remark >- $\mathrm{MA}(\aleph_{0})$ is true - this is known as the [[Rasiowa–Sikorski lemma]]. > - $\mathrm{MA}(2^{\aleph_0})$ is false: $[0, 1]$ is a [[compact space|compact]] [[Hausdorff space]], which is [[separable space|separable]] and so ccc. It has no [[isolated point]]s, so points in it are nowhere dense, but it is the union of $2^{\aleph_0}=\mathfrak c$ many points. (See the equivalent condition below.) **Martin's axiom** $\mathrm{MA}$: For every $\kappa<\mathfrak{c}$, $\mathrm{MA}(\kappa)$ holds. # Equivalent forms of $\mathrm{MA}(\kappa)$ The following statements are equivalent to $\mathrm{MA}(\kappa)$: * If $X$ is a compact Hausdorff [[topological space]] that satisfies the [[Countable chain condition|ccc]] then $X$ is not the union of $\leq \kappa$ [[nowhere dense]] subsets. * If $P$ is a non-empty upwards ccc [[Partially ordered set|poset]] and $Y$ is a family of cofinal subsets of $P$ with $|Y|\leq \kappa$ then there is an upwards-directed set $A$ such that $A$ meets every element of $Y$. * Let $A$ be a non-zero ccc [[Boolean algebra (structure)|Boolean algebra]] and $F$ a family of subsets of $A$ with $|F|\leq \kappa$ Then there is a boolean homomorphism $φ: A → Z/2Z$ such that for every $X$ in $F$ either there is an $a$ in $X$ with $φ(a) = 1$ or there is an upper bound $b$ for $X$ with $φ(b) = 0$. # Consequences * The union of $\kappa$ or fewer [[null set]]s in an atomless σ-finite [[Borel measure]] on a [[Polish space]] is null. In particular, the union of $\kappa$ or fewer subsets of $\mathbb{R}$ of [[Lebesgue measure]] 0 also has Lebesgue measure 0. * A compact Hausdorff space $X$ with $|X|<2^{\kappa}$ is [[Compact space|sequentially compact]], i.e., every sequence has a convergent subsequence. * No non-principal [[ultrafilter]] on $\mathbb{N}$ has a base of cardinality lt;\kappa$. * ??Equivalently for any $x$ in $β\mathbb{N} \\ \mathbb{N}$ we have 𝜒($x$) ≥ 𝛋, where 𝜒 is the [[character (topology)|character]] of $x$, and so 𝜒($\beta\mathbb{N}$) ≥ 𝛋. * $\mathrm{MA}(\aleph_{1})$ implies that a product of ccc topological spaces is ccc (this in turn implies there are no [[Suslin line]]s). * MA + ¬CH implies that there exists a Whitehead group that is not free; [[Saharon Shelah|Shelah]] used this to show that the [[Whitehead problem]] is independent of ZFC. * $\mathrm{MA}(\aleph_{1})$ implies that every [[Trees#^aron|Aronszajn tree]] is [[Trees#^spec|special]]. In particular there are no [[Trees#^souslin|Souslin trees]]. ^spec-aron # Further development Martin's axiom has generalizations called the [[Proper forcing axiom]] and [[Martin's maximum]]. Sheldon W. Davis has suggested in his book that Martin's axiom is motivated by the [[Baire category theorem]].