1.1 Topological spaces

Topological spaces form the foundation of modern geometry, providing a framework to study continuity and nearness without relying on distance. They generalize metric spaces, allowing for more abstract notions of proximity and connectedness.

Open and closed sets, bases, and subbases are key concepts in topological spaces. These structures define the topology and enable the study of continuous functions, homeomorphisms, and important properties like compactness and connectedness in a wide range of mathematical contexts.

Definition of topological spaces

• A topological space is a set $X$ together with a collection $\tau$ of subsets of $X$, called open sets, satisfying certain axioms
• The open sets in a topological space form a topology on the set, which encodes the notion of nearness or proximity between points without relying on a metric or distance function
• The axioms that define a topology ensure that the collection of open sets is closed under arbitrary unions and finite intersections, and that both the empty set and the entire space are open

Open sets in topological spaces

• A subset $U$ of a topological space $(X, \tau)$ is called open if it belongs to the topology $\tau$
• Open sets are the fundamental building blocks of a topological space and are used to define various topological properties
• Examples of open sets in common topological spaces:
  • In the standard topology on $\mathbb{R}$, open intervals $(a, b)$ and their unions are open sets
  • In the discrete topology on any set $X$, every subset of $X$ is open

Closed sets in topological spaces

• A subset $F$ of a topological space $(X, \tau)$ is called closed if its complement $X \setminus F$ is open
• Closed sets are the complements of open sets and play a crucial role in the study of topological spaces
• Examples of closed sets in common topological spaces:
  • In the standard topology on $\mathbb{R}$, closed intervals $[a, b]$ and their intersections are closed sets
  • In the discrete topology on any set $X$, every subset of $X$ is closed

Basis for a topology

• A basis for a topology on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$ such that every open set in the topology can be expressed as a union of elements from $\mathcal{B}$
• A basis generates a unique topology on $X$ by taking all possible unions of its elements
• Examples of bases for common topological spaces:
  • The collection of open intervals $(a, b)$ forms a basis for the standard topology on $\mathbb{R}$
  • The collection of all singleton sets $\{x\}$ forms a basis for the discrete topology on any set $X$

Subbasis for a topology

• A subbasis for a topology on a set $X$ is a collection $\mathcal{S}$ of subsets of $X$ such that the collection of all finite intersections of elements from $\mathcal{S}$ forms a basis for the topology
• A subbasis generates a unique topology on $X$ by first forming a basis from finite intersections of its elements and then taking all possible unions of the basis elements
• Example of a subbasis for a common topological space:
  • The collection of all open intervals of the form $(-\infty, a)$ and $(b, \infty)$ forms a subbasis for the standard topology on $\mathbb{R}$

Hausdorff spaces

• A topological space $(X, \tau)$ is called a Hausdorff space (or T2 space) if for any two distinct points $x, y \in X$, there exist disjoint open sets $U, V \in \tau$ such that $x \in U$ and $y \in V$
• Hausdorff spaces are a fundamental class of topological spaces that ensure a certain level of separation between distinct points
• Many important topological spaces, such as metric spaces and manifolds, are Hausdorff spaces

Continuous functions between topological spaces

• A function $f: (X, \tau_X) \to (Y, \tau_Y)$ between two topological spaces is called continuous if the preimage of every open set in $Y$ is open in $X$
• Continuous functions preserve the topological structure and are the primary objects of study in topology
• Composition of continuous functions is continuous, and the identity function is always continuous

Homeomorphisms of topological spaces

• A function $f: (X, \tau_X) \to (Y, \tau_Y)$ between two topological spaces is called a homeomorphism if it is a continuous bijection with a continuous inverse
• Homeomorphic spaces are considered topologically equivalent, as they have the same topological properties
• Examples of homeomorphic spaces:
  • The open interval $(0, 1)$ and the real line $\mathbb{R}$ are homeomorphic
  • The unit circle $S^1$ and the one-point compactification of $\mathbb{R}$ are homeomorphic

Topological properties preserved by homeomorphisms

• Homeomorphisms preserve many important topological properties, such as:
  • Compactness
  • Connectedness
  • Hausdorff property
  • Countability axioms (first and second countability, separability, Lindelöf property)
• If a topological property is preserved by homeomorphisms, it is called a topological invariant

Subspaces of topological spaces

• A subset $A$ of a topological space $(X, \tau)$ can be endowed with a subspace topology, making it a topological space in its own right
• The subspace topology on $A$ is defined as $\tau_A = \{U \cap A : U \in \tau\}$, consisting of the intersections of open sets in $X$ with $A$

Subspace topology

• The subspace topology is the coarsest topology on $A$ that makes the inclusion map $i: A \hookrightarrow X$ continuous
• Many topological properties, such as compactness and connectedness, are inherited by subspaces from the ambient space
• Examples of subspaces with the subspace topology:
  • The closed interval $[0, 1]$ as a subspace of $\mathbb{R}$ with the standard topology
  • The unit circle $S^1$ as a subspace of $\mathbb{R}^2$ with the standard topology

Closure and interior in subspaces

• The closure of a subset $B$ of a subspace $A$ in the subspace topology is given by $\overline{B}^A = \overline{B} \cap A$, where $\overline{B}$ is the closure of $B$ in the ambient space $X$
• The interior of a subset $B$ of a subspace $A$ in the subspace topology is given by $\text{int}_A(B) = \text{int}(B) \cap A$, where $\text{int}(B)$ is the interior of $B$ in the ambient space $X$
• These properties allow for the study of topological concepts within subspaces and their relationship to the ambient space

Product topology

• The product topology on the Cartesian product of topological spaces is a natural way to define a topology that is compatible with the topologies of the individual spaces
• Given a family of topological spaces $\{(X_i, \tau_i)\}_{i \in I}$, the product topology on $\prod_{i \in I} X_i$ is generated by the basis consisting of products of open sets from each factor space

Finite product topology

• For a finite product of topological spaces $X_1 \times X_2 \times \cdots \times X_n$, the product topology is generated by the basis consisting of sets of the form $U_1 \times U_2 \times \cdots \times U_n$, where each $U_i$ is open in $X_i$
• The product topology on a finite product is the coarsest topology that makes all projection maps continuous
• Example of a finite product space:
  • The torus $T^2 = S^1 \times S^1$ with the product topology

Infinite product topology

• For an infinite product of topological spaces $\prod_{i \in I} X_i$, the product topology is generated by the basis consisting of sets of the form $\prod_{i \in I} U_i$, where each $U_i$ is open in $X_i$ and $U_i = X_i$ for all but finitely many indices $i$
• The infinite product topology is also known as the Tychonoff topology
• Example of an infinite product space:
  • The Hilbert cube $[0, 1]^{\mathbb{N}}$ with the product topology

Projection maps in product spaces

• For each index $j \in I$, the projection map $\pi_j: \prod_{i \in I} X_i \to X_j$ is defined by $\pi_j((x_i)_{i \in I}) = x_j$
• Projection maps are always continuous with respect to the product topology
• Projection maps play a crucial role in the study of product spaces and their relationship to the factor spaces

Continuity in product spaces

• A function $f: Y \to \prod_{i \in I} X_i$ is continuous with respect to the product topology if and only if each component function $\pi_i \circ f: Y \to X_i$ is continuous
• This characterization simplifies the study of continuity in product spaces by reducing it to continuity in the factor spaces
• Example of a continuous function in a product space:
  • The diagonal map $\Delta: X \to X \times X$ defined by $\Delta(x) = (x, x)$ is continuous with respect to the product topology

Quotient topology

• The quotient topology is a way to define a topology on a set that is obtained by identifying or "gluing together" certain points in a given topological space
• Given a topological space $(X, \tau)$ and an equivalence relation $\sim$ on $X$, the quotient topology on the quotient set $X/\sim$ is the finest topology that makes the quotient map $q: X \to X/\sim$ continuous

Quotient maps

• A surjective function $q: (X, \tau_X) \to (Y, \tau_Y)$ is called a quotient map if the topology $\tau_Y$ is the quotient topology induced by $q$
• Quotient maps are characterized by the property that a subset $U \subseteq Y$ is open in $Y$ if and only if $q^{-1}(U)$ is open in $X$
• Quotient maps are a powerful tool for constructing new topological spaces from existing ones

Continuity of quotient maps

• Quotient maps are always continuous by definition
• A function $f: (Y, \tau_Y) \to (Z, \tau_Z)$ is continuous if and only if the composition $f \circ q: (X, \tau_X) \to (Z, \tau_Z)$ is continuous, where $q: (X, \tau_X) \to (Y, \tau_Y)$ is a quotient map
• This property allows for the study of continuity in quotient spaces by lifting functions to the original space

Examples of quotient spaces

• The torus $T^2$ can be obtained as a quotient space of the square $[0, 1] \times [0, 1]$ by identifying opposite sides
• The projective plane $\mathbb{RP}^2$ can be obtained as a quotient space of the sphere $S^2$ by identifying antipodal points
• The Klein bottle can be obtained as a quotient space of the square $[0, 1] \times [0, 1]$ by identifying opposite sides with a twist

Connectedness in topological spaces

• Connectedness is a fundamental topological property that captures the idea of a space being in one piece
• A topological space $(X, \tau)$ is connected if it cannot be expressed as the union of two disjoint non-empty open sets

Connected spaces

• Examples of connected spaces:
  • The real line $\mathbb{R}$ with the standard topology
  • The unit circle $S^1$ with the subspace topology
  • Any interval $[a, b]$ with the subspace topology
• Examples of disconnected spaces:
  • The set $\{0, 1\}$ with the discrete topology
  • The set $\mathbb{Q}$ of rational numbers with the subspace topology inherited from $\mathbb{R}$

Path-connected spaces

• A topological space $(X, \tau)$ is path-connected if for any two points $x, y \in X$, there exists a continuous function $\gamma: [0, 1] \to X$ such that $\gamma(0) = x$ and $\gamma(1) = y$
• Path-connectedness is a stronger property than connectedness, as every path-connected space is connected, but the converse is not true
• Example of a space that is connected but not path-connected:
  • The topologist's sine curve $\{(x, \sin(1/x)) : 0 < x \leq 1\} \cup \{(0, y) : -1 \leq y \leq 1\}$

Components and path components

• The (connected) components of a topological space $(X, \tau)$ are the maximal connected subsets of $X$
• The path components of a topological space $(X, \tau)$ are the maximal path-connected subsets of $X$
• Every path component is contained in a unique component, but a component may contain multiple path components

Local connectedness

• A topological space $(X, \tau)$ is locally connected if for every point $x \in X$ and every open set $U$ containing $x$, there exists a connected open set $V$ such that $x \in V \subseteq U$
• Local connectedness is a property that relates the global notion of connectedness to the local structure of the space
• Examples of locally connected spaces:
  • The real line $\mathbb{R}$ with the standard topology
  • The unit circle $S^1$ with the subspace topology

Compactness in topological spaces

• Compactness is a fundamental topological property that generalizes the notion of closed and bounded subsets in Euclidean spaces
• A topological space $(X, \tau)$ is compact if every open cover of $X$ has a finite subcover

Open cover of a topological space

• An open cover of a topological space $(X, \tau)$ is a collection $\{U_i\}_{i \in I}$ of open sets such that $X = \bigcup_{i \in I} U_i$
• A subcover of an open cover $\{U_i\}_{i \in I}$ is a subcollection $\{U_j\}_{j \in J}$, where $J \subseteq I$, that still covers $X$
• Compactness can be characterized by the property that every open cover has a finite subcover

Compact spaces

• Examples of compact spaces:
  • Any finite topological space
  • The closed interval $[0, 1]$ with the subspace topology
  • The unit circle $S^1$ with the subspace topology
• Examples of non-compact spaces:
  • The real line $\mathbb{R}$ with the standard topology
  • The open interval $(0, 1)$ with the subspace topology

Compact subspaces

• A subset $K$ of a topological space $(X, \tau)$ is compact if it is compact as a subspace with the subspace topology
• Closed subsets of compact spaces are compact (Closed subset property)
• Compact subsets of Hausdorff spaces are closed (Closed in Hausdorff property)

Continuity and compactness

• The continuous image of a compact space is compact (Continuous image property)
• If $f: (X, \tau_X) \to (Y, \tau_Y)$ is a continuous function and $K \subseteq X$ is compact, then $f(K)$ is compact in $Y$
• This property is useful for proving the compactness of spaces constructed using continuous functions

Tychonoff's theorem for product spaces

• Tychonoff's theorem states that the product of any collection of compact spaces is compact with respect to the product topology
• This theorem is a powerful tool for constructing compact spaces and proving the compactness of certain function spaces
• Example of an application of Tychonoff's theorem:
  • The Hilbert cube $[0, 1]^{\mathbb{N}}$ is compact with respect to the product topology

Separation axioms

• Separation axioms are properties that describe how well points and closed sets can be separated by open sets in a topological space
• The separation axioms form a hierarchy, with each axiom implying the previous ones

T0, T1, T2 (Hausdorff) spaces

• A topological space $(X, \tau)$ is T0 (Kolmogorov) if for any two distinct points $x, y \in X$, there exists an open set containing one of the points but not the other
• A topological space $(X, \tau)$ is T1 (Fréchet) if for any two distinct points $x, y \in X$, there exist open sets $U, V$ such that $$x \in