Real and Complex Analysis MAT311
Exercise Sheet Five
Monday 18 March 2024
1 Real Analysis
These questions are from the prescribed textbook [Ros86]
1.1 Metric Spaces
Definition 1.1. Interior point: Let S be a subset of the metric space M = (X, d). A point
p ∈ S is called an interior point of S if there is an open ball in M with centre p and sufficiently
small radius for it to be contained in subset S.
Definition 1.2. Interior: Let S be a subset of the metric space M = (X, d). The set of
interior points of S is called the interior of S, denoted by either int S or S o . If S is open, then
S o = S. Otherwise, S o ⊂ S.
Question 1. Prove that the set of interior points of S is an open subset of M, called the
interior of S, that contains all other open subsets of M that are contained in S.
Definition 1.3. Closure: Let S be a subset of the metric space M = (X, d). The closure of
S, denoted S, is the intersection of all closed subsets M that contain S.
Question 2. Show that
(a) S ⊃ S, and S is closed if and only if S = S
(b) S is the set of all limits of sequences of points of S that converge in M
(c) A point p ∈ X is in S if and only if any ball in M of centre p contains points of S, which
is true if and only if p is not an interior point of S c .
Definition 1.4. Boundary: Let S be a subset of the metric space M = (X, d). The boundary
of S, denoted ∂S, is the intersection of S and is complement in M. I.e.
∂S = S ∩ S c .
Question 3. Show that
(a) M is the disjoint union of the interior of S, the interior of S c and the boundary of S
(b) S is closed if and only if S contains its boundary ∂S
(c) S is open if and only if S and its boundary ∂S are disjoint.
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Exercise Sheet Five
Monday 18 March 2024
1 Real Analysis
These questions are from the prescribed textbook [Ros86]
1.1 Metric Spaces
Definition 1.1. Interior point: Let S be a subset of the metric space M = (X, d). A point
p ∈ S is called an interior point of S if there is an open ball in M with centre p and sufficiently
small radius for it to be contained in subset S.
Definition 1.2. Interior: Let S be a subset of the metric space M = (X, d). The set of
interior points of S is called the interior of S, denoted by either int S or S o . If S is open, then
S o = S. Otherwise, S o ⊂ S.
Question 1. Prove that the set of interior points of S is an open subset of M, called the
interior of S, that contains all other open subsets of M that are contained in S.
Definition 1.3. Closure: Let S be a subset of the metric space M = (X, d). The closure of
S, denoted S, is the intersection of all closed subsets M that contain S.
Question 2. Show that
(a) S ⊃ S, and S is closed if and only if S = S
(b) S is the set of all limits of sequences of points of S that converge in M
(c) A point p ∈ X is in S if and only if any ball in M of centre p contains points of S, which
is true if and only if p is not an interior point of S c .
Definition 1.4. Boundary: Let S be a subset of the metric space M = (X, d). The boundary
of S, denoted ∂S, is the intersection of S and is complement in M. I.e.
∂S = S ∩ S c .
Question 3. Show that
(a) M is the disjoint union of the interior of S, the interior of S c and the boundary of S
(b) S is closed if and only if S contains its boundary ∂S
(c) S is open if and only if S and its boundary ∂S are disjoint.
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