Topology I: Topological Spaces and the Fundamental Group
MATH 131

Fall 2012
0.1. METRIC SPACES
0.1
1
Metric spaces
The definition of a topological space is one of the great achievements of 20th
century mathematics. It encodes exactly the information we need about any
geometric object in order to to study those of its properties t
Topology I: Topological Spaces and the Fundamental Group
MATH 131

Fall 2012
Math 131: Topology I
Final
Practice Problems
Emily Riehl
Conventions: Capital letters X, Y denote (nonempty) topological spaces. Disjoint unions,
products, subspaces, quotients, and gluings are assumed to be given their usual topologies
unless explicitly
Topology I: Topological Spaces and the Fundamental Group
MATH 131

Fall 2012
ON THE CONSTRUCTION OF NEW TOPOLOGICAL SPACES
FROM EXISTING ONES
EMILY RIEHL
Abstract. In this note, we introduce a guiding principle to dene topologies
for a wide variety of spaces built from existing topological spaces. The topologies soconstructed wil
Topology I: Topological Spaces and the Fundamental Group
MATH 131

Fall 2012
1
10/12/2012
Recall from last time the following denition
Denition 1.1 (Disconnectedness). X is disconnected if an open, non
=
empty disjoint U and V where U V X .
By homework we will know that X is disconnected if and only if it has a
proper clopen subsp
Topology I: Topological Spaces and the Fundamental Group
MATH 131

Fall 2012
Math 131: Topology I
Midterm 2
October 31, 2012
Emily Riehl
Conventions: Capital letters X, Y denote topological spaces. Disjoint unions, products,
subspaces, quotients, and gluings are assumed to be given their usual topologies unless
explicitly stated o
Topology I: Topological Spaces and the Fundamental Group
MATH 131

Fall 2012
Math 131: Topology I
Midterm 2
Practice Problems
Emily Riehl
Conventions: Capital letters X, Y denote (nonempty) topological spaces. Disjoint unions,
products, subspaces, quotients, and gluings are assumed to be given their usual topologies
unless explic
Topology I: Topological Spaces and the Fundamental Group
MATH 131

Fall 2012
Math 131: Topology I
Midterm 1
Practice Problems
Emily Riehl
Conventions: Capital letters A, B, X, Y, Z denote topological spaces. Disjoint unions, products, subspaces, quotients, and gluings are assumed to be given their usual topologies unless explicitl
Topology I: Topological Spaces and the Fundamental Group
MATH 131

Fall 2012
Math 131: Topology I
Midterm 1
October 3, 2012
Emily Riehl
Conventions: Capital letters A, B, X, Y, Z denote topological spaces. Disjoint unions, products, subspaces, quotients, and gluings are assumed to be given their usual topologies unless explicitly
Topology I: Topological Spaces and the Fundamental Group
MATH 131

Fall 2012
Math 131: Topology I
Sample solutions
Emily Riehl
1.I.1. Let p1 < p2 < . . . be all prime numbers (regardless of whether there is a nite
or innite number of them). Recall that Spec(Z) = cfw_ p1 , p2 , . . ., and the Zariski topology
Z on Spec(Z) has close