Every basis set is open in . So we can ask it the topology is different than the Euclidean topology generated by all open intervals in . Show that there are intervals of the form that are open in that are not open sets in .
Let , and let and . Both and are topologies in , but every element in is also an element in . Then this happens we say that is a weaker topology than .Exercise 4 provides an example. More formally,
If a topology on contains all single point sets, then every subset is open and our our topology is the discrete topology. Also, if a topology on contains all two-point sets, then if ,, and are in it follows that is in the topology and we again have the.discrete topology. Consider the topology
The only sets not in are and , but adding either set to will produce the discrete topology. So is a strongest topology possible other than the discrete topology.
Let . Are there any topologies on such that is not the discrete topology but there are no stronger topologies on other than the discrete topology? Explain.
Let . Are there any topologies on such that is not the indiscrete topology but there are no weaker topologies on other than the indiscrete topology? Explain.
In general, there may be many different bases for a given topology, and two different bases can have the same cardinality. This is not the case for finite topological space. Let be a finite set and let be a topology on . In this exercise we will show that there is a minimal basis for the topology . That is, there is a basis min of such that if is any other basis for , then min .
Below is a list of distinct topologies on . Each topology lies in one or more sequences of topologies ordered by coarseness. For each topology , list the longest sequence(s) of topologies that start , ordered by coarseness.
Let be two subsets in a topological space . What can you say about the relationships between IntInt and IntIntIntInt, respectively? Verify your results.
Let be a nonempty set and let be an element in . Let be the collection of subsets of consisting of ,, and all of the subsets of that contain . Show that is a topology on . (This topology is called the particular point topology).
Let be a nonempty set and let be an element in . Let be the collection of subsets of consisting of ,, and all of the subsets of that do not contain . Show that is a topology on . (This topology is called the excluded point topology.)
One application of topology is in digital image displays, such as a computer screen. A digital image display is a rectangular array of pixels and can be modeled using a digital plane. In this exercise we consider a simplification of the digital plane β the digital line β which we consider as an infinite length one-dimensional collection of pixels. For each we define
Show that the collection is a basis for a topology on . (The resulting topology is called the digital line topology .β7β The digital line models a one-dimensional array of pixels, where the even integers are the pixels and the odd integers are boundaries between the pixels. Information about the digital plane can be found in Chapter 20.)
Explain why the Zariski topology when is just the cofinite topology on . That is, show that every set that is open in the cofinite topology is open in the Zariski topology and that every set that is open in the Zaariski topology is open in the cofinite topology.
For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.
If and are topologies on a space , then is also a topology on .
This digital line topology has applications in digital processing β see Introduction to Topology: Pure and Applied by Colin Adams and Robert Franzosa , Pearson Education, Inc., 2008, Sections 1.4 and 11.3. The set with the digital line topology is called the digital line.
