1.4 Subspaces and product spaces

Subspaces and product spaces are key concepts in topology, allowing us to build new spaces from existing ones. Subspaces zoom in on parts of a space, while product spaces combine multiple spaces into one.

These ideas help us understand complex topological structures by breaking them down or building them up. They're essential tools for exploring how spaces relate to each other and for creating new spaces with specific properties.

Subspace Topology

Defining Subspace Topology

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• Subspace topology induces topology on a subset of a topological space from the original topology
• For topological space $(X, T)$ and subset $A$ of $X$, subspace topology on $A$ consists of intersections of open sets in $X$ with $A$
• Preserves open sets of original space within subset context
• Collection of open sets in subspace topology forms basis for topology on subset
• Closed sets in subspace topology intersect closed sets in original space with subset
• Represents coarsest topology on subset making inclusion map continuous
• Also known as relative topology, emphasizing relationship to original space

Properties and Examples of Subspace Topology

• Inherits topological properties from parent space
• Subspace of a metric space inherits metric structure
• Real line with standard topology induces subspace topology on intervals (open, closed, or half-open)
• Complex plane subspace topology on unit circle preserves circular structure
• Subspace topology on rational numbers $\mathbb{Q}$ as subset of real numbers $\mathbb{R}$ with standard topology
• Three-dimensional Euclidean space $\mathbb{R}^3$ induces subspace topology on planes, lines, and surfaces within it
• Discrete topology on a set induces discrete subspace topology on any subset

Subspace as Topological Space

Proving Subspace is a Topological Space

• Demonstrate subspace satisfies three axioms of topological space
• Show empty set and entire subset are open in subspace topology
• Prove union of any collection of open sets in subspace topology remains open
• Demonstrate intersection of finite collection of open sets in subspace topology stays open
• Use subspace topology definition to relate openness in subspace to original space
• Employ set operation properties, particularly distributivity of intersection over union
• Conclude by showing subspace topology satisfies all conditions of topological space

Examples and Applications

• Unit interval $[0,1]$ as subspace of real line with standard topology
• Sphere $S^2$ as subspace of $\mathbb{R}^3$ with Euclidean topology
• Cantor set as subspace of real line with standard topology
• Open disk in complex plane as subspace of $\mathbb{C}$ with standard topology
• Subspace proof technique applies to various topological constructions (quotient spaces, identification spaces)
• Subspaces preserve many topological properties (compactness, connectedness, separability)
• Understanding subspaces crucial for analyzing topological manifolds and their submanifolds

Product Topology

Defining Product Topology

• Product topology defined on Cartesian product of topological spaces
• For spaces $(X, T_X)$ and $(Y, T_Y)$, product topology on $X \times Y$ generated by basis of products $U \times V$, where $U$ open in $X$ and $V$ open in $Y$
• Represents coarsest topology making all projection maps continuous
• Open sets in product topology are unions of basis elements (rectangles with open sides)
• Generalizes to finite or infinite number of factor spaces
• Closed sets in product topology more complex, not necessarily products of closed sets in factor spaces
• Satisfies universal property for products in category of topological spaces

Examples and Properties of Product Topology

• Real plane $\mathbb{R}^2$ as product of two real lines with standard topology
• Torus as product of two circles $S^1 \times S^1$
• Cylinder as product of circle and real line $S^1 \times \mathbb{R}$
• Infinite-dimensional cube $[0,1]^\infty$ as product of countably many unit intervals
• Product topology on function spaces (e.g., space of continuous functions with pointwise convergence topology)
• Hilbert cube as product of countably many closed intervals
• Product topology preserves many topological properties (Hausdorff, regularity, normality) for finite products

Properties of Product Spaces

Projection Maps and Continuity

• Projection maps from product space to factor spaces continuous in product topology
• Product topology coarsest topology making all projection maps continuous
• Box topology finest topology making projection maps continuous
• Box topology coincides with product topology for finite products, differs for infinite products
• Continuous functions into product spaces characterized by continuity of compositions with projection maps
• Examples of projection maps: $\pi_1: \mathbb{R}^2 \to \mathbb{R}$ given by $\pi_1(x,y) = x$
• Projection maps used to define sections and graphs of functions between topological spaces

Topological Properties of Product Spaces

• Compactness, connectedness, and separability of product spaces depend on factor space properties
• Tychonoff theorem states product of any collection of compact spaces compact in product topology
• Product of connected spaces connected, but product of path-connected spaces may not be path-connected for infinite products
• Separability of product spaces: countable product of separable spaces separable (Hewitt-Marczewski-Pondiczery theorem)
• Metrizability of product spaces: countable product of metrizable spaces metrizable
• Product of first-countable spaces not necessarily first-countable (e.g., uncountable product of real lines)
• Dimension of product spaces: product of n-dimensional spaces has dimension at most sum of individual dimensions