Topological Spaces

Course Material

Lecture 11 - 16/07Lecture 12 - 17/07Lecture 13 - 23/07Lecture 14 - 24/07Lecture 15 - 30/07Lecture 15 - 30/07Lecture 16 - 31/07

1. A topological space is a set equipped with a topology - a family of subsets satisfying the topology axioms.
2. Every metric space is a topological space.
3. There exist topological spaces that are not metric spaces.
4. A subset of a topological space can be given the subspace topology.
5. The interior of a set \(Y\), denoted \(\mathrm{Int}(Y)\) is open in the topological space.
6. A topological space is the most general setting in which convergence of sequences can be defined.
7. Continuous maps between topological spaces are defined using the preimage formulation.
8. The preimage definition is equivalent to the traditional definition in topological spaces.
9. Some topological spaces are Hausdorff.
10. The defining feature of a Hausdorff space is the uniqueness of limits.
11. There are topological spaces where limits are unique, yet the space is not Hausdorff.

16/07 Lecture Slides

1. In \(\mathbb{R}\) with the standard topology, every open set is a countable union of disjoint open intervals.
2. This property does not hold in \(\mathbb{R}^2\).
3. To describe topologies using simpler families of subsets, we use the concepts of a base for a topology and a local base for neighbourhoods.
4. The topology itself forms a base, but it is often possible to reduce the base to a smaller family.
5. In a metric space, the family of open balls with rational radii forms a base.
6. Any family of subsets satisfying conditions (1) and (2) on page 41 forms a base for a topology.
7. Definitions of continuity and convergence can be stated in terms of a base.
8. A base can be further reduced to a subbase; any family of subsets can serve as a subbase for a topology.
9. The resulting topology is the smallest topology containing the given family of subsets.
10. Convergence of sequences can be tested using only the elements of a subbase.

17/07 Lecture Slides

23/07 Lecture Slides

24/07 Lecture Slides

30/07 Lecture Slides

30/07 Lecture Slides

31/07 Lecture Slides

Introduction and Definitions

A topological space is a set \(X\) together with a set of subsets \(\tau = \mathcal{O}(X) \subseteq \mathcal{P}(X)\) satisfying

1. \(\emptyset, X \in \mathcal{O}(X)\)
2. If \({V_i}_{i \in I} \subseteq \mathcal{O}(X)\), then \(\bigcup_{i \in I} V_i \in \mathcal{O}(X)\).
3. If \(V_1, V_2 \in =\mathcal{O}(X)\), then \(V_1 \cap V_2 \in \mathcal{O}(X)\).

Common Examples of Topologies

1. Let \((X, d)\) be a metric space. Then we have already seen the metric topology \(\tau_d\).
2. Let \(X\) be any set. The coarse topology is \(\tau = \{\emptyset, X\}\).
3. Let \(X\) be any set. The discrete topology is \(\tau = \mathcal{P}(X)\).
4. Let \(X\) be any set. The cofinite topology is \(\tau = \{Y \subseteq X : Y^C \text{ is finite}\} \cup \{\emptyset\}\).
5. Let \(X\) be any set. The cocountable topology is \(\tau = \{Y \subseteq X: Y^C \text{ is countable}\} \cup \{\emptyset\}\).

Let \((X, \tau)\) be a topological space, and let \(Y \subseteq X\). The subspace topology (also called relative topology) is \(\tau|_Y = \{V \cap Y : V \in \tau\}\).

Let \((X, \tau)\) be a topological space. A subset \(Y \subseteq X\) is closed if \(Y^C\) is open.

Let \((X, \tau)\) be a topological space.

1. Let \(x \in X\) be a point. An open neighbourhood of \(x\) is a set \(V \in \tau\) such that \(x \in V\). A neighbourhood of \(x\) is any set containing an open neighbourhood of \(x\). We will denote the collection of neighbourhoods of \(x\) by \(\mathrm{Nbhd}(x)\).
2. Let \(Y \subseteq X\) be a subset. The interior of \(Y\) is

\[ \mathrm{Int}(Y) = \{y \in E: \exists V_y \in \mathrm{Nbhd}(y) \text{ such that } V_y \subseteq Y\}. \]

Let \((X, \tau)\) be a topological space. For any subset \(Y \subseteq X\), the interior \(\mathrm{Int}(Y)\) is an open set.

Let \((X, \tau)\) be a topological space, and let \(Y \subset X\).

1. The boundary is \(\mathrm{Bd}(Y) = X \setminus (\mathrm{Int}(Y) \sqcup \mathrm{Int}(Y^C))\)
2. The closure is \(\mathrm{cl}(Y) = \mathrm{Int}(Y) \cup \mathrm{Bd}(Y)\).

Convergence in Topological Spaces

Let \((X, \tau)\) be a topological space. A sequence \(\{x_n\}_{n = 1}^\infty \subseteq X\) converges to \(x \in X\) if for every \(V \in \mathrm{Nbhd}(x)\), there is a \(K(V) \in \mathbb{N}\) such that \(x_n \in V\) when \(n \geq K\).

Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces. A function \(f: X \to Y\) is continuous if for every \(V \in \tau_Y\), we have \(f^{-1}(V) \in \tau_X\).

Let \((X, \tau_X), (Y, \tau_Y)\) and \((Z, \tau_Z)\) be topological spaces. If \(f: X \to Y\) and \(g: Y \to Z\) are continuous functions, then \((g \circ f): X \to Z\) is also continuous.

A topological space \(X\) has the Hausdorff property if for every pair of distinct points \(x, y \in X\), there are neighbourhoods \(V(x, y) \in \mathrm{Nbhd}(x)\) and \(U(x, y) \in \mathrm{Nbhd}(y)\) such that \(V(x, y) \cap U(x, y) = \emptyset\).

A sequence in a Hausdorff space has at most one limit.

Bases for Topologies

Let \((X, \tau)\) be a topological space.

1. A base for \(\tau\) is a subset \(\mathcal{B} \subset \tau\) such that every \(V \in \tau\) can be expressed as a union of elements of \(\mathcal{B}\):

   \[V = \bigcup_{i \in I} V_i, \text{ where } V_i \in \mathcal{B}, \forall i \in I \]

2. A local base for \(\tau\) at a point \(x \in X\) is a collection \(\mathcal{LB}_x \subseteq \tau\) of open neighbourhoods of \(x\) such that if \(U\) is any neighbourhood of \(x\), there is a \(V \in \mathcal{LB}_x\) such that \(V \subseteq U\).

Theorem: Let \(X\) be a set, and let \(\mathcal{B} \subseteq \mathcal{P}(X)\) be a collection of subsets. Then

\[\tau = \{ V \subseteq X: V \text{ is a union of sets in } \mathcal{B} \} \]

is a topology iff the following conditions hold:

1. \(\bigcup_{V \in \mathcal{B}} V = X\) ("\(\mathcal{B}\) covers \(X\)")
2. for every \(V_1\) and \(V_2\) in \(\mathcal{B}\) and every \(x \in V_1 \cap V_2\), there is \(V \in \mathcal{B}\) such that \(x \in V \subseteq V_1 \cap V_2\).

Let \(X\) be a set, and \(S \subseteq \mathcal{P}(X)\) be a collection of subsets. Define \(\mathcal{B}\) to be the set of all finite intersections of sets in \(S\).

\[\mathcal{B} = \{V_1 \cup \cdots \cup V_n : V_k \in S, k = 1, \dots, n \}.\]

(We allow the empty intersection \(X\)). Then \(\mathcal{B}\) satisfies the conditions for a base in the previous theorem, so

\[\tau(S) = \{V \subseteq X, V \text{ is a union of sets in } \mathcal{B}\}\]

is a topology. We call \(S\) a subbase for \(\tau(S)\), and say that \(\tau\) is generated by \(S\).

Pointwise and Weak Convergence

Let \(X\) be a set and let \(Y = F(X, \mathbb{R})\). For each \(x \in X, y \in \mathbb{R}\), and \(\epsilon > 0\), define

\[V_{x, y, \epsilon} = \{g \in Y : |g(x) - y| < \epsilon \}.\]

Then let

\[S = \{V_{x, y, \epsilon} \}_{x \in X, y \in \mathbb{R}, \epsilon > 0}.\]

Finally, define the topology of pointwise convergence to be \(\tau_{pt} = \tau(S)\), the topology generated by the subbase \(S\).

Theorem: A sequence of functions \(f_n : X \to \mathbb{R}\) converges pointwise to \(f\) if and only if \(f_n \to f\) in the topology \(\tau_{pt}\).

Let \(\mathbf{H}\) be a Hilbert space, such as \(\mathbb{R}^n\) or \(\ell^2\). We say that a sequence of vectors \(\{ x_n \}_{n=1}^\infty\) converges weakly to a vector \(\mathbf{x} \in \mathbf{H}\) if for every vector \(\mathbf{y} \in \mathbf{H}\), we have

\[\langle \mathbf{x}_n, \mathbf{y} \rangle \to \langle \mathbf{x}, \mathbf{y} \rangle.\]

Let \((a, b)\) be an open interval in \(\mathbb{R}\). A sequence of functions is said to converge compactly if it converges uniformly on every closed subinterval \([c, d] \subseteq (a, b)\).

Countability

A topological space \((X, \tau)\) is said to be:

1. First countable if every point in \(X\) has a countable local base for \(\tau\).
2. Second countable if \(X\) has a countable base for \(\tau\).

A topological space is separable if it contains a countable dense subset.

Let \((X, \tau)\) be a topological space. A local base \(\{V_n\}_{n \in \mathbb{N}}\) at a point \(x\) is called nested if \(V_n \subseteq C_m, \forall n \geq m\).

Let \((X, \tau)\) be a first countable topological space. Then a subset \(Y \subseteq X\) is closed if and only if for every sequence in \(Y\) which converges (in \(X\)), the limit is in \(Y\).

Nets

A directed set is a set \(\Lambda\), together with a binary relation \(\leq\) satisfying, for all \(i, j, k \in Lambda\)

1. \(i \leq i\)
2. \(i \leq j\) and \(j \leq k \implies i \leq k\)
3. \(\exists m \in \Lambda\) such that \(i, j \leq m\).

A net is a function from a directed set \((\Lambda, \leq)\) to a set \(X\). As for sequences, we'll use the notation \(\{x_\lambda\}_{\lambda \in \Lambda}\).

Let \((X, \tau)\) be a topological space. A net \(\{x_\lambda\}_{\lambda \in \Lambda}\) converges to a point \(x \in X\) if for every neighbourhood \(V\) of \(x\), there is an \(\alpha(V) \in \Lambda\) such that \(x_\lambda \in V\) when \(\lambda \geq \alpha\).

Theorem: Let \((X, \tau)\) be a topological space. The \(Y \subseteq X\) is closed if and only if \(Y\) contains the limits of all of its convergent nets.

Comparison of Topologies

Suppose \(\tau\) and \(\sigma\) are two topologies on a set \(X\) such that \(\tau \subseteq \sigma\). Then we say that \(\tau\) is coarser and \(\sigma\) is finer.

Homeomorphisms

Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces. A bijection \(f: X \to Y\) is a called a homeomorphism if both \(f\) and \(f^{-1}\) are continuous. We say that \(X\) and \(Y\) are homeomorphic if there exists a homomorphism \(f: X \to Y\).

Two metric \((X, d_X)\) and \((Y, d_Y)\) are said to be isometric if there is a bijection \(f: X \to Y\) such that

\[_Y(f(x_1), f(x_2)) = d_X(x_1, x_2), \forall x_1, x_2 \in X.\]

Two metric spaces \((X, d_X)\) and \((Y, d_Y)\) are said to be equivalent if there is a bijection \(f: X \to Y\) and constants \(k, K > 0\) such that

\[k \cdot d_X(x_1, x_2) \leq d_Y(f(x_1), f(x_2)) \leq K \cdot d_X(x_1, x_2), \forall x_1, x_2 \in X.\]

Two normed spaces \((X, \lVert \cdot \rVert_X)\) and \((Y, \lVert \cdot \rVert_Y)\) are said to be equivalent if there is a bijection \(f: X \to Y\) and constants \(k, K > 0\) such that

\[k \cdot \lVert \mathbf{x} \rVert_X \leq \lVert f(\mathbf{x}) \rVert_Y \leq K \cdot \lVert \mathbf{x} \rVert_X, \forall x_1, x_2 \in X.\]

Homeomorphism Invariants

A topological space \((X, \tau)\) is connected if it is not the union of two disjoint nonempty open subsets. A subset \(Y \subseteq X\) is connected if it is connected in the subspace topology.

Theorem: The interval \([0, 1]\) is connected.

Theorem: Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces, and suppose \(f: X \to Y\) is a continuous function. If \(X\) is connected, then so is \(f(X)\) (as a subset of \(Y\)).

Lemma: Let \((X, \tau)\) be a topological space, and suppose that \(\{W_i\}_{i \in I}\) are connected subsets such that \(\bigcap_{i \in I} W_i \neq \emptyset\). Then \(\bigcap_{i \in I} W_i\) is connected.

The equivalence classes of \(X\) under \(~^{ct}\) are called the connected components of \(X\).

A topological space \((X, \tau)\) is path-connected if for every pair of points \(x, y \in X\), there is a continuous function \(f: [0, 1] \to X\) such that \(f(0) = x\) and \(f(1) = y\). A subset \(Y \subseteq X\) is path-connected if it is path-connected in the subspace topology.

Theorem: If a topological space is path-connected, then it is connected.

Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces. The product topology on \(X \times Y\) is the topology generated by the base

\[\{U \times V\}_{U \in \tau_X, V \in \tau_Y}.\]

Let \(\{(X_i, \tau_i)\}_{i \in I}\) be a collection of topological spaces. The box topology \(\prod_{i \in I} X_i\) is the topology with base

\[\mathcal{B} = \left\{\prod_{i \in I} U_i : U_i \in \tau_i, \forall i \in I \right\}\]

Let \(\{(X_i, \tau_i)\}_{i \in I}\) be a set of topological spaces. The product topology on \(\prod_{i \in I} X_i\) is the topology generated by the base

\[\left\{ \prod_{i \in I} U_i : U_i \in \tau_i, \forall i \in I \text{ and } U_i = X_i \text{ for all but finitely many } i \in I \right\}\]