About the coherent topology wiki page

In summary, the inclusion maps for a topological union are topological embeddings if and only if the generating spaces satisfy the compatibility condition and X_i \cap X_j is closed in X_i for each i,j. This is not always true for CW-complexes, as there are counterexamples where the inclusion of a n-skeleton into the CW-complex is not a topological embedding.
  • #1

quasar987

Science Advisor
Homework Helper
Gold Member
4,807
32
On the wiki page on coherent topology, and more precisely, topological union (aka topology generated by a collection of spaces) (http://en.wikipedia.org/wiki/Coherent_topology#Topological_union), it is said that if the generating spaces {X_i} satisfy the compatibility condition that for each i,j, the subspace topologies induced on [itex]X_i\cap X_j[/itex] by X_i and X_j are the same, then the inclusion maps [itex]\iota_i:X_i\rightarrow \cup_iX_i[/itex] are topological embeddings (i.e. homeomorphisms onto their images).

Is this true? I tried proving it but without success but could not find a counter example either.

For instance, CW-complexes are the topological union of their n-skeletons. Is it always true that the inclusion of a n-skeleton into the CW-complex is a topological embedding?
 
Last edited:
Physics news on Phys.org
  • #2
It's not true; in the case of CW complexes, you need an additional condition that [itex]X_i \cap X_j[/itex] is closed in [itex]X_i[/itex] for each i, j. Counterexample (due to my officemate Scott Van Thuong):

Take a triangle [itex]\Delta ABC[/itex], and let [itex]X_1 = \overline{AB}[/itex], [itex]X_2 = \overline{BC}[/itex], and [itex]X_3 = \overline{CA}[/itex]. Give [itex]X_1[/itex] the Euclidean topology, and give both [itex]X_2[/itex] and [itex]X_3[/itex] the indiscrete topology. Let [itex]X[/itex] denote the whole triangle with the coherent topology. Now take an open neighborhood [itex]U[/itex] in [itex]X_1[/itex] with [itex]A \in U[/itex] but [itex]B \not \in U[/itex]. If the inclusion [itex]X_1 \hookrightarrow X[/itex] were an embedding, there would exist some open set [itex]V \subseteq X[/itex] with [itex]U = V \cap X_1[/itex]. Since V contains B, it must contain all of [itex]X_2[/itex]; therefore [itex]C \in V[/itex]. Since V contains C, it must contain all of [itex]X_3[/itex]; in particular, it contains A. But this contradicts the assumption.
 
Last edited:

What is a coherent topology?

A coherent topology is a mathematical concept that describes the structure of a set in terms of its open sets and their relationships. It is a way of defining the "closeness" of points in a set, and is used in topology, which is a branch of mathematics that studies the properties of spaces and their transformations.

How is a coherent topology defined?

A coherent topology is defined by a collection of open sets that satisfy certain properties. These properties include: (1) the empty set and the entire set must be open; (2) the intersection of any finite number of open sets is also open; and (3) the union of any collection of open sets is also open.

What are some examples of coherent topologies?

Some common examples of coherent topologies include the discrete topology, the indiscrete topology, and the Euclidean topology. The discrete topology defines every subset of a set as open, the indiscrete topology defines only the empty set and the entire set as open, and the Euclidean topology defines open sets as those that can be expressed as an open ball around each point.

What is the significance of coherent topologies?

Coherent topologies are important in mathematics because they allow us to define and study the properties of spaces in a precise and rigorous way. They are also useful in applications, such as in physics and computer science, where spaces and their transformations are studied and utilized.

How does a coherent topology relate to other mathematical concepts?

Coherent topologies are closely related to concepts such as continuity, connectedness, and compactness. They are also used in the study of metric spaces, which are sets with a defined notion of distance between points. In addition, coherent topologies are connected to algebraic structures, such as groups and rings, through the use of topological spaces.

Suggested for: About the coherent topology wiki page

Replies
20
Views
2K
Replies
2
Views
2K
Replies
5
Views
2K
Replies
3
Views
2K
Replies
9
Views
369
Back
Top