In this paper we explore the extent to which the algebraic structure of a monoid
M
M
determines the topologies on
M
M
that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or
T
1
T_1
topologies; Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids and inverse monoids.
If
M
M
is a topological monoid such that every homomorphism from
M
M
to a second countable topological monoid
N
N
is continuous, then we say that
M
M
has automatic continuity. We show that many well-known, and extensively studied, monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid
N
N
\mathbb {N} ^\mathbb {N}
; the full binary relation monoid
B
N
B_{\mathbb {N}}
; the partial transformation monoid
P
N
P_{\mathbb {N}}
; the symmetric inverse monoid
I
N
I_{\mathbb {N}}
; the monoid
Inj
(
N
)
\operatorname {Inj}(\mathbb {N})
consisting of the injective transformations of
N
\mathbb {N}
; and the monoid
C
(
2
N
)
C(2^{\mathbb {N}})
of continuous functions on the Cantor set
2
N
2^{\mathbb {N}}
.
The monoid
N
N
\mathbb {N} ^\mathbb {N}
can be equipped with the product topology, where the natural numbers
N
\mathbb {N}
have the discrete topology; this topology is referred to as the pointwise topology. We show that the pointwise topology on
N
N
\mathbb {N} ^\mathbb {N}
, and its analogue on
P
N
P_{\mathbb {N}}
, is the unique Polish semigroup topology on these monoids. The compact-open topology is the unique Polish semigroup topology on
C
(
2
N
)
C(2 ^\mathbb {N})
, and on the monoid
C
(
[
0
,
1
]
N
)
C([0, 1] ^\mathbb {N})
of continuous functions on the Hilbert cube
[
0
,
1
]
N
[0, 1] ^\mathbb {N}
. The symmetric inverse monoid
I
N
I_{\mathbb {N}}
has at least 3 Polish semigroup topologies, but a unique Polish inverse semigroup topology. The full binary relation monoid
B
N
B_{\mathbb {N}}
has no Polish semigroup topologies, nor do the partition monoids. At the other extreme,
Inj
(
N
)
\operatorname {Inj}(\mathbb {N})
and the monoid
Surj
(
N
)
\operatorname {Surj}(\mathbb {N})
of all surjective transformations of
N
\mathbb {N}
each have infinitely many distinct Polish semigroup topologies.
We prove that the Zariski topologies on
N
N
\mathbb {N} ^\mathbb {N}
,
P
N
P_{\mathbb {N}}
, and
Inj
(
N
)
\operatorname {Inj}(\mathbb {N})
coincide with the pointwise topology; and we characterise the Zariski topology on
B
N
B_{\mathbb {N}}
.
Along the way we provide many additional results relating to the Markov topology, the small index property for monoids, and topological embeddings of semigroups in
N
N
\mathbb {N}^{\mathbb {N}}
and inverse monoids in
I
N
I_{\mathbb {N}}
.
Finally, the techniques developed in this paper to prove the results about monoids are applied to function clones. In particular, we show that: the full function clone has a unique Polish topology; the Horn clone, the polymorphism clones of the Cantor set and the countably infinite atomless Boolean algebra all have automatic continuity with respect to second countable function clone topologies.