Skip to main content

Topology

Sam Macdonald

Section 2.1 Topology and Continuity

“Topology is precicely the mathematical discipline that allows passage from local to global.”
―Rene Thom

Subsection 2.1.1 Topology

Definition 2.1.1.

A topology on a set \(X\) is a collection \(\cT \) of subsets of \(X\) satisfying:
  1. \(\es \) and \(X\) are elements of \(\cT \text{;}\)
  2. whenever \(A_\alpha \) is in \(\cT \) for all \(\alpha \text{,}\) then \(\cup_\alpha A_\alpha \) is in \(\cT \text{;}\) and
  3. whenever \(A_1,\dots,A_n\) are in \(\cT \text{,}\) then \(\cap_{i=1}^n A_i\) is in \(\cT \text{.}\)

Definition 2.1.2. .

A topological space is a set \(X\) together with a topology \(\cT \) on \(X\text{,}\) often denoted \((X,\cT )\text{.}\) The elements of \(X\) are called points in \(X\text{,}\) and the elements of \(\cT \) are called open sets in \(X\) or open sets relative to \(X\) or open sets rel \(X\text{.}\)

Example 2.1.3. Examples of Topologies and Topological Spaces:.

  1. Indiscrete topology on a set \(X: \cT_{\text{ind}} := \{\es, X\}\text{.}\)
  2. Discrete topology on a set \(X: \cT_{\text{dis}} := \{U | U \sse X\} = \cP (X) =\) the set of all subsets of \(X\text{.}\)
  3. Euclidean topology on \(\R^n: \cT_{\text{Eucl}} := \{U \sse \R | \text{ for all } p \in U \text{ there is an } r > 0 \text{and} q \in \R^n \text{such that} p \in B(q,r) \sse U \}\text{,}\) where \(B(q,r) = B_n(q,r) := \{x \in \R^n | d_{\text{Eucl}}(q,x) < r\}\) is the ball of radius \(r\) centered at \(q\) (and \(d_Eucl\) is Euclidean distance).
  4. Euclidean topology relative to a subset \(X \sse \R^n: \cT_{\text{Eucl sub}} := \{U \cap X | U \sse \R^n \text{ and } U\in\cT_{\text{Eucl}}\}\text{.}\)
  5. Finite complement topology on a set \(X: \cT_{fc} := \{U \sse X | X\setminus U \text{ is a finite set }\} \cup \{\es \}\text{.}\)
  6. Included point topology on a set \(X\) with included point \(a \in X: \cT_{\text{incl}} := \{U \sse X | a \in U\} \cup \{\es \}\text{.}\)
  7. Excluded point topology on a set \(X\) with excluded point \(a \in X: \cT_{\text{excl}} := \{U \sse X | a \not\in U\} \cup \{X\}\text{.}\)
  8. Infinite ray topology on \(\R : \cT_{\text{ray}} := \{ (-∞,a) | a \in \R \} \cup \{\es,\R \}\text{.}\)
  9. Preimage topology constructed on a set \(X\) from a function \(f:X \to Y\) and a topology \(\cT_Y\) on \(Y: \cT_f := \{f\inv(U) | U \in \cT_Y\}\text{.}\)
  10. Line with two origins with set \((\R \setminus \{0\}) \cup \{*,@\}\) and topology \(\cT := \{U | U \sse \R,\sm \{0\} \text{ and }U \in \cT_{\text{Eucl}}\} \cup \{(U\setminus \{0\}) \cup \{*\} | U \sse \R, 0 \in U, \text{ and }U \in \cT_{\text{Eucl}}\} \cup \{(U\setminus \{0\}) \cup \{@\} | U \sse \R, 0 \in U, \text{ and }U \in \cT_{\text{Eucl}}\} \cup \{(U\setminus \{0\}) \cup \{*,@\} | U \sse \R, 0 \in U, \text{ and }U \in \cT_{\text{Eucl}}\}\text{.}\)
  11. On the finite set \(X = \{a,b,c\}: \cT = \{ \es, \{a\}, \{a,b\}, \{a,c\}, X\}\text{.}\)

Remark 2.1.4.

Definition 2.1.6. Courser and Finer Topologies.

Let \(\cT \) and \(\cT'\) be topologies on a set \(X\text{.}\) If \(\cT \sse \cT \text{,}\) then the topology \(\cT \) is called smaller or coarser than \(\cT'\text{,}\) and the topology \(\cT'\) is called larger or finer than \(\cT \text{.}\)

Example 2.1.7.

Subsection 2.1.2 Continuity

Motivation: Unraveling the \(\varepsilon -\delta \) definition of continuous in Euclidean space to an equivalent definition on open sets.

Definition 2.1.8. Continuous Function.

Let \(X\) and \(Y\) be topological spaces, and let \(f: X \to Y\) be a function. The function \(f\) is continuous if and only if for every open set \(U\) in \(Y\text{,}\) the preimage \(f\inv(U)\) is open in \(X\text{.}\)

Example 2.1.9.

Examples

Example 2.1.12.

Definition 2.1.14. Homeomorphism.

Let \(X\) and \(Y\) be topological spaces. A function \(f: X \to Y\) is a homeomorphism if \(f\) is a continuous bijection and \(f\inv\) is also continuous. The spaces \(X\) and \(Y\) are called homeomorphic, or topologically equivalent, if there is a homeomorphism from \(X\) to \(Y\text{;}\) in symbols this is written \(X \cong Y\text{.}\)

Big picture:.

Euclidean, spherical and hyperbolic metric spaces of the same dimension are homeomorphic. \(E^2 \cong H^2 \cong S^2\setminus \{(0,0,1)\}\text{.}\)
Definition 2.1.16. Homeomorphism Problem.
The Homeomorphism Problem asks: Is there a computer algorithm that, upon input of two topological spaces \(X\) and \(Y\text{,}\) can determine whether or not \(X \cong Y\text{?}\)
(Backing up a step: How do you input a space into a computer?!)
Example 2.1.17.

Subsection 2.1.3 Bases

Definition 2.1.18. Basis.

A basis on a set \(X\) is a collection \(\cB \) of subsets of \(X\) satisfying:
  1. For each \(p\) in \(X\) there is an \(N_p\) in \(\cB \) such that\(p \in N_p\text{,}\) and
  2. whenever \(N_1,N_2\) are in \(\cB \) and \(p\) is an element of \(N_1 \cap N_2\text{,}\) then there is an \(N_3\) in \(\cB \) such that \(p \in N_3 \sse N_1 \cap N_2\text{.}\)

Definition 2.1.19. Topology Generated by a Basis.

The topology generated by a basis \(\cB \) is \(\cT (\cB ) := \{U \sse X | \text{ for all }p\in U \text{ there is an } N\in \cB \text{ such that }p \in N \sse U\}\text{.}\)

Example 2.1.21.

Example 2.1.23. The Euclidean Basis.

The Euclidean basis on \(\R^n\) is \(\cB_{\text{Eucl}} := \{ B_n(q,r) | q \in \R^n \text{ and }r > 0\}\text{.}\)

Definition 2.1.25. Lower Limit Topology.

The lower limit topology on the set \(\R \) is the topology \(\cT ll := \cT (\cB ll)\) generated by the basis \(\cB ll := \{[a,b) | a,b \in \R, a < b\}\text{.}\)

Definition 2.1.28. Subbasis.

A subbasis on a set \(X\) is a collection \(\cS \) of subsets of \(X\) satisfying \(\cup_{U \in \cS } U = X\text{.}\)

Definition 2.1.29.

The basis generated by a subbasis \(\cS\) is \(\cB (\cS ) := \{\cap i=1n Ui | n \in \N \text{ and }Ui \in \cS \text{ for all }i\}\text{.}\) The topology generated by a subbasis \(\cS\) is \(\cT (\cS ) := \cT (\cB (\cS ))\text{.}\)

Example 2.1.30.

<Prev^TopNext>