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
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