A brief note about notation: this document was coded with a proposed HTML math switch which browsers do not read quite yet. Since the proposed codes are nearly identical to the corresponding TeX-codes, perhaps this won't be such a big problem. The lower- and upper-case Greek letters are pretty easy to spot, but beginners might not recognize a few others:

Throughout this summary we assume $X$ to be Tychonoff, we let $\beta X$ represent the Stone-Cech compactification of $X$, and we let $\upsilon X$ represent its Hewitt-Nachbin realcompactification.

The citations provided are not necessarily the original sources for the definitions.

A nice source for additional background information on realcompact spaces is [W].

[GJ,5.15] $X$ is realcompact iff every ultrafilter of zero-sets with the countable intersection property is fixed.

[F] $X$ is almost realcompact iff every ultrafilter of regular closed sets with the countable intersection property is fixed.

[D] $X$ is a-realcompact iff every ultrafilter of closed sets with the countable intersection property is fixed.

Before we define pure spaces, we'll need the following:

A family $E\; =\; \{E$_n: n ∈ ω} of non-empty subsets of a space $X$ is called an interlacing on $X$ iff $\cup E$ is a cover of $X$ and for each $n\; \in \; \omega$, $U\; \in \; E$_n, $U$ is open in $\cup \; E$_n. An interlacing $E$ is $\delta$-suspended from a family $H$ of subsets of a space $X$ iff for arbitrary $n\; \in \; \omega$, $x\; \in \; \cup \; E$_n , there exists a countable subcollection $F$ of $H$ such that $st(x,E$_n) ∩ (∩ F) is empty.

[A] A space $X$ is called pure iff for each free closed ultrafilter $F$ on $X$, there is an interlacing which is $\delta$-suspended from $F$.

Before we define neat spaces, we'll need a bit of notation.

For an ultrafilter $H$, $\lambda (H)\; =\; min\{|F|:F\; \subseteq \; H$ and $\cap F$ is empty}.

[Sak] A space $X$ is called neat iff for every free closed ultrafilter $H$ with the countable intersection property on $X$, there is a system $[X$_γ,U_γ,f_γ] with $\gamma \; \in \; \Gamma$ such that

1. and $\cup \{X$_γ: γ ∈ Γ} = X,
2. for each $\gamma \; \in \; \Gamma$, $U$_γ is an open collection in $X$ and $X$_γ ⊆ ∪U_γ,
3. each $f$_γ mapping $X$_γ to $U$_γ is such that if $A\; \in \; [X]\le \omega$ and $f$_γ|_A is injective, then $cl$_∪Uγ ⊆ ∪{f_γ(x): x ∈ A},
4. for each $\gamma \; \in \; \Gamma$ and _γ, $\exists G\; \in \; H$ such that $f$_γ(x) ∩ X_γ ∩ G is empty.

[Mil] A space $X$ is called $\delta$-neat iff for every free closed ultrafilter $H$ with the countable intersection property on $X$, there is a system $[X$_n,U_n,f_n] with $n\; \in \; \omega$ such that

1. $\cup \{X$_n: n ∈ ω} = X,
2. for each $n\; \in \; \omega$, $U$_n is an open collection in $X$ and $X$_n ⊆ ∪U_n,
3. each $f$_n mapping $X$_n to $U$_n is such that if $A\; \in \; [X]\le \omega$ and $f$_n|_A is injective, then $cl$_∪Un ⊆ ∪{f_n(x): x ∈ A},
4. for each $n\; \in \; \omega$ and $x\; \in \; X$_n, $\exists G\; \in \; H$ such that $f$_n(x) ∩ X_n ∩ G is empty.

[Mro] $X$ is realcompact iff for each point $p\; \in \; \beta X\; -\; X$, there exists a continuous function $f$ on $\beta X$ such that $f(p)=0$ and $f$ is positive on $X$.

[D] $X$ is c-realcompact iff for each point $p\; \in \; \beta X\; -\; X$, there exists a normal lower semicontinuous function $f$ on $\beta X$ such that $f(p)=0$ and $f$ is positive on $X$.

[SS] $X$ is p-realcompact iff every zero-set of $\beta X$ that meets $\beta X\; -\; X$ meets $\beta X\; -\; \upsilon X$.

If $C$_ρ (X) represents the collection of continuous functions with realcompact support, then $X$ is realcompact iff $C$_ρ(X) = C(X).

For any set , let M^A = $\{f\; \in \; C(X):\; A\; \subseteq \; cl$_{β X}Z(f)} and O^A = $\{f\; \in \; C(X):\; A\; \subseteq \; int$_{β X}cl_{β X}Z(f)}. Let $C$_K(X) denote the ring of continuous functions with compact support, $C$_ψ(X) the ring of functions with pseudocompact support, and $I(X)$ the intersection of all free maximal ideals of $C(X)$. Then

[M] $X$ is $\mu$-compact iff $C$_K(X)=I(X).

[M] $X$ is $\psi$-compact iff $C$_K(X)=C_ψ(X).

[JM] $X$ is $\eta$-compact iff $C$_ψ(X)=I(X).

[BS] $X$ is $\lambda$-compact iff O^{$\upsilon \; X\; -\; X$} = M^{$\upsilon \; X\; -\; X$}

[Shi][GJ,15.20] $X$ is realcompact iff $X$ admits a compatible complete uniformity and contains no closed discrete subspace of measurable cardinality.

[E,8.5.13] We call a space completely uniformizable (or Dieudonne complete or topologically complete) iff it admits a compatible complete uniformity.

[Bac][BLS,3.9] A space is said to be hyperisocompact [strongly isocompact] (isocompact) iff every relatively pseudocompact [strongly relatively pseudocompact] (countably compact) closed subset is compact.

[GJ,8.5] $X$ is realcompact iff $X\; =\; \upsilon X$.

[BvD] $X$ is nearly realcompact iff $\beta X\; -\; \upsilon X$ is dense in $\beta X\; -\; X$.

[GP,14.3] $X$ is realcompact iff each 2-valued Baire measure in $X$ is $\tau$-additive.

[GP,8.1] A space $X$ is called weakly Borel-complete iff each 2-valued Borel measure in $X$ is weakly $\tau$-additive.

[GP,8.1] $X$ is closed-complete (or $\alpha$-realcompact) iff each 2-valued regular Borel measure in $X$ is $\tau$-additive.

References