Metric and Hilbert Spaces
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updated: 4 November 2014
Lecture 5: Topological spaces, interiors and closures
A topological space is a set with a collection of subsets of such that
(a) |
and
|
(b) |
If then
|
(c) |
If and
then
|
Let be a topological space.
An open set of is
A closed set of is a subset such that is open.
Let be a set. The discrete topology on is
(the "power set of
Let be a metric space. Let and
let The
ball of radius at is the set
The metric space topology on is
Let be a topological space. Let
The subspace topology on is
Let and
be topological spaces. The product of the sets and is the set
The product topology on is
Examples of open and closed sets
Let with the metric given by
and the metric space topology. Then
(think of a door that is not open and not closed, i.e. ajar) and
Let be a topological space and let
A neighbourhood of
is a subset such that there exists an open set of with
and
The neighbourhood filter of is
Let be a topological space and let
The interior of is the subset of such that
(a) |
is open and and
|
(b) |
if is open and then
|
The closure of is the subset of such that
(a) |
is closed and and
|
(b) |
if is closed and then
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In English:
is the largest open set contained in and
is the smallest closed set containing
An interior point of is a point such that there exists a neighbourhood
of such that
A close point of is a point such that if is a neighbourhood of
then
Let be a topological space. Let
(a) |
The interior of is the set of interior points of
|
(b) |
The closure of is the set of close points of
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![](plus.gif) |
![](minus.gif) |
Proof of (a). |
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Let
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To show: |
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To show: |
(aa) |
|
(ab) |
|
|
(aa) |
Let
Then there exists a neighbourhood of with
So there exists an open set with
Since and is open
So
So
|
(ab) |
To show: If then
Assume
Then is open and
So is an interior point of
So
So
So
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|
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Notes and References
These are a typed copy of Lecture 5 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on August 5, 2014.
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