We work throughout in the theory
Z
F
\mathsf {ZF}
with the axiom of determinacy,
A
D
\mathsf {AD}
. We introduce and prove some club uniformization principles under
A
D
\mathsf {AD}
and
A
D
R
\mathsf {AD}_\mathbb {R}
. Using these principles, we establish continuity results for functions of the form
Φ
:
[
ω
1
]
ω
1
→
ω
1
\Phi \colon [{\omega _{1}}]^{\omega _{1}} \rightarrow {\omega _{1}}
and
Ψ
:
[
ω
1
]
ω
1
→
ω
1
ω
1
\Psi \colon [{\omega _{1}}]^{\omega _{1}} \rightarrow {}^{\omega _{1}}{\omega _{1}}
. Specifically, for every function
Φ
:
[
ω
1
]
ω
1
→
ω
1
\Phi \colon [\omega _1]^{\omega _1} \rightarrow \omega _1
, there is a club
C
⊆
ω
1
C \subseteq \omega _1
so that
Φ
↾
[
C
]
∗
ω
1
\Phi \upharpoonright [C]^{\omega _1}_*
is a continuous function. This has several consequences such as establishing the cardinal relation
|
[
ω
1
]
>
ω
1
|
>
|
[
ω
1
]
ω
1
|
|[{\omega _{1}}]^{>{\omega _{1}}}| > |[{\omega _{1}}]^{\omega _{1}}|
and answering a question of Zapletal by showing that if
⟨
X
α
:
α
>
ω
1
⟩
\langle X_\alpha : \alpha > \omega _1\rangle
is a collection of subsets of
[
ω
1
]
ω
1
[\omega _1]^{\omega _1}
with the property that
⋃
α
>
ω
1
X
α
=
[
ω
1
]
ω
1
\bigcup _{\alpha > \omega _1}X_\alpha = [\omega _1]^{\omega _1}
, then there is an
α
>
ω
1
\alpha > \omega _1
so that
X
α
X_\alpha
and
[
ω
1
]
ω
1
[\omega _1]^{\omega _1}
are in bijection.
We show that under
A
D
R
\mathsf {AD}_\mathbb {R}
everywhere
[
ω
1
]
>
ω
1
[\omega _1]^{>\omega _1}
-club uniformization holds which is the following statement: Let
c
l
u
b
ω
1
\mathsf {club}_{\omega _1}
denote the collection of club subsets of
ω
1
\omega _1
. Suppose
R
⊆
[
ω
1
]
>
ω
1
×
c
l
u
b
ω
1
R \subseteq [\omega _1]^{>\omega _1} \times \mathsf {club}_{\omega _1}
is
⊆
\subseteq
-downward closed in the sense that for all
σ
∈
[
ω
1
]
>
ω
1
\sigma \in [\omega _1]^{>\omega _1}
, for all clubs
C
⊆
D
C \subseteq D
,
R
(
σ
,
D
)
R(\sigma ,D)
implies
R
(
σ
,
C
)
R(\sigma ,C)
. Then there is a function
F
:
d
o
m
(
R
)
→
c
l
u
b
ω
1
F \colon {\mathrm {dom}}(R) \rightarrow \mathsf {club}_{\omega _1}
so that for all
σ
∈
d
o
m
(
R
)
\sigma \in {\mathrm {dom}}(R)
,
R
(
σ
,
F
(
σ
)
)
R(\sigma ,F(\sigma ))
.
We show that under
A
D
\mathsf {AD}
almost everywhere
[
ω
1
]
>
ω
1
[{\omega _{1}}]^{>{\omega _{1}}}
-club uniformization holds which is the statement that for every
R
⊆
[
ω
1
]
>
ω
1
×
c
l
u
b
ω
1
R \subseteq [{\omega _{1}}]^{>{\omega _{1}}} \times \mathsf {club}_{\omega _{1}}
which is
⊆
\subseteq
-downward closed, there is a club
C
C
and a function
F
:
d
o
m
(
R
)
∩
[
C
]
∗
>
ω
1
→
c
l
u
b
ω
1
F \colon {\mathrm {dom}}(R) \cap [C]^{>{\omega _{1}}}_* \rightarrow \mathrm {club}_{\omega _{1}}
so that for all
σ
∈
d
o
m
(
R
)
∩
[
C
]
∗
>
ω
1
\sigma \in {\mathrm {dom}}(R) \cap [C]^{>{\omega _{1}}}_*
,
R
(
σ
,
F
(
σ
)
)
R(\sigma ,F(\sigma ))
.