. Supposeγ : C → Γ ⊆ℝ^{2}is one to one onto and continuous. Then ℝ^{2}∖Γ consists of two components, a bounded component(called the inside) U_{i}and an unbounded component (called the outside), U_{o}. Also the boundary ofeach of these two components of ℝ^{2}∖ Γ is Γ and Γ has empty interior.

Proof: First, Γ has empty interior by invariance of domain theorem. By the Jordan separation
theorem,

2
ℝ ∖Γ = Uo ∪ Ui

where these on the right are the connected components of the set on the left, both open sets. Only one of
them is unbounded. This is U_{o} and U_{i} is the bounded component. Thus Γ ∪U_{i}∪U_{o} = ℝ^{2}. Let ∂U ≡U∖U
for U either U_{o} or U_{i}. If x ∈ Γ, and is not a limit point of U_{i}, then there is B

(x,r)

which contains no
points of U_{i}. If it is not a limit point of U_{o} either, then adjusting r we could get a ball having points of
neither U_{i} nor U_{o} which would produce a ball contained in Γ which, as noted above, can’t
happen. Hence every x ∈ Γ is a limit point of either U_{i} or U_{o}. Thus Γ ⊆ ∂U_{i}∪ ∂U_{o}. If you have
x ∈ ∂U_{i}, then it is a limit point of U_{i}. If it is not in Γ, then it would be in U_{o} and consequently
could not be a limit point of U_{i}. Thus ∂U_{i}⊆ Γ. Similarly ∂U_{o}⊆ Γ and so ∂U_{i}∪ ∂U_{o} = Γ,
U_{i}∪U_{o} = ℝ^{2}.

Suppose ∂U_{i}∖ ∂U_{o}≠∅. Say p ∈ ∂U_{i}∖ ∂U_{o}. Then there is a ball B

(p, r)

which has empty intersection
with ∂U_{o}. If for every r this ball has points of U_{o} then p would be a limit point of U_{o}. Hence p ∈ ∂U_{o}
which is assumed to not take place. Hence there is a ball B

(p, r)

which has empty intersection with U_{o}. It
follows that this ball is contained in U_{i}. Let S be those points x of Γ for which B

(x,δ)

is contained in U_{i}
for some δ > 0. Thus S is the intersection of Γ with an open set. Indeed, if x ∈ S, then if

∥y− x∥

< δ for y ∈ Γ, then B

(y,δ − ∥y− x∥)

⊆U_{i}. Hence, if you let

ˆΓ

be what is left of Γ
after deleting S, it follows that

ˆΓ

is a compact subset of Γ. It corresponds to Ĉ a compact
subset of C. By the Jordan separation theorem,

ˆΓ

^{C} is a single open connected set. This is
because this is obviously the case for Ĉ, being the part of a circle which is off an open set.
Now let x ∈ U_{o} and y ∈ U_{i}. Then there is a continuous curve from x to y which misses every
point of

Γˆ

. However, it must hit some point of Γ because U_{i},U_{o} are different components.
Therefore, it contains a point p of S. Let p be the first point of Γ which this curve intersects.
Then this point must be in ∂U_{o}. But B

(p,r)

is contained in U_{i} for some r > 0 and so this
ball cannot contain any points of U_{o}. But this is impossible if p is in ∂U_{o}. This contradiction
shows that ∂U_{i}∖ ∂U_{o} = ∅. Thus ∂U_{i}⊆ ∂U_{o}. Similarly ∂U_{o}⊆ ∂U_{i}. Hence Γ = ∂U_{i} = ∂U_{o}.
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