Notes by Jerzy Wojciechowski
1 Topological Spaces
1.1 Metric Spaces
1.1.1 Convergence of sequences of real numbers.
1.1.2 Continuity of real functions.
1.1.3 Definition of a metric space.
Remark.
1.1.4 Example (Euclidean spaces).
1.1.5 Example (Hilbert space).
1.1.6 Example (integration metric).
1.1.7 Example (supremum metric).
1.1.8 Definition of continuity.
1.1.9 Definition of convergence of sequences.
1.1.10 Open balls.
1.1.11 Open sets.
1.1.12 Theorem (properties of metric spaces).
1.1.13 Theorem (continuity for metric spaces).
1.1.14 Homework 1 (due 1/14).
Problem 1.
Problem 2.
Problem 3.
Problem 4.
1.2 Topologies
1.2.1 Example of a convergence not induced by a metric.
1.2.2 Definition of topology.
Examples.
1.2.3 Comparing topologies on the same set.
1.2.4 Closed sets.
1.2.5 Proposition (properties of closed sets).
Example.
1.2.6 Homework 2 (due 1/21)
1.2.7 Neighborhoods.
1.2.8 Proposition (properties of nbhds).
1.2.9 Proposition (topology from nbhds).
1.2.10 Homework 3 (due 1/28).
1.3 Derived Concepts
1.3.1 Interior
Remarks.
1.3.2 Proposition (properties of interior).
1.3.3 Closure.
1.3.4 Proposition (properties of closure).
1.3.5 Theorem (characterization of closure).
1.3.6 Limit points.
1.3.7 Theorem (closure and derived set).
Corollary.
1.3.8 Boundary.
1.3.9 Theorem (closure and boundary).
1.3.10 Isolated points.
1.3.11 Perfect sets.
1.3.12 Example (the Cantor set).
1.3.13 Dense sets.
1.3.14 Homework 4 (due 2/4).
1.4 Bases
1.4.1 Proposition (family of topologies).
1.4.2 Subbases.
1.4.3 Proposition (topology from subbasis).
1.4.4 Linear order
1.4.5 Order topology
1.4.6 Homework 5 (due 2/11).
1.4.7 Well-ordered sets.
1.4.8 Well-ordering principle.
1.4.9 The well-ordered set \(\bra {0,\ome }\) as a topological space.
1.4.10 Theorem (the space \(\bra {0,\ome }\)).
1.4.11 Bases.
1.4.12 Theorem (basis for a topology).
1.4.13 Homework 6 (due 2/18).
1.4.14 Proposition (comparing topologies).
1.4.15 Equivalent metrics.
1.4.16 Proposition (equivalent metrics).
1.4.17 Corollary (bounded metric)
1.4.18 Local basis.
1.4.19 Proposition (properties of nbhd basis).
1.4.20 Theorem (topology from nbhd basis).
1.4.21 Homework 7 (due 2/25).
1.5 Subspaces
1.5.1 Proposition (subspace topology).
1.5.2 Relative topology.
1.5.3 Proposition (closed sets in subspaces).
1.5.4 Proposition (relative metric induces relative topology).
1.5.5 Proposition (closed and open subspaces).
1.5.6 Proposition (relative subbasis, basis and nbhd basis).
1.5.7 Proposition (relative closure and derived set).
1.5.8 Proposition (relative interior and boundary).
1.5.9 Proposition (relative linear order).
1.5.10 Homework 8 (due 3/3)
2 Continuity and the Product Topology
2.1 Continuous Functions
2.1.1 Definition of a continuous function.
2.1.2 Theorem (characterization of continuity).
2.1.3 Theorem (continuity and basis).
2.1.4 Theorem (composition of continuous functions).
2.1.5 Theorem (characterization of subspace topology).
2.1.6 Localized continuity.
2.1.7 Theorem (localized continuity).
2.1.8 Theorem (Gluing Lemma).
2.1.9 Locally finite family.
2.1.10 Proposition (closure and locally finite family).
2.1.11 Corollary (closure and locally finite family).
2.1.12 Homeomorphism.
2.1.13 Open and closed functions.
2.1.14 Theorem (characterization of homeomorphisms).
2.1.15 Proposition (characterization of closed functions).
2.1.16 Theorem (characterization of open functions).
2.1.17 Topological embedding.
2.1.18 Homework 9 (due 4/7)
2.2 Product Spaces
2.2.1 Proposition (basis for product topology)
2.2.2 Product topology.
2.2.3 Proposition (characterization of product topology).
2.2.4 Proposition (basis for product topology from bases).
2.2.5 Proposition (product topology on \(\br ^{2}\)).
2.2.6 Theorem (product and subspace topologies commute).
2.2.7 Theorem (continuity into products).
2.2.8 Infinite Cartesian products.
2.2.9 Box topology.
2.2.10 Product topology.
2.2.11 Proposition (characterization of infinite products).
2.2.12 Proposition (subbasis for infinite product).
2.2.13 Proposition (basis for infinite product).
2.2.14 Proposition (infinite products and subspaces commute).
2.2.15 Theorem (infinite products and continuity).
2.2.16 Theorem (countable products are metrizable).
2.2.17 Metrizable spaces.
2.2.18 Exercises.
3 Connectedness
3.1 Connected Spaces
3.1.1 Separation.
3.1.2 Definition of connected spaces.
3.1.3 Theorem (connectedness and functions into discrete).
3.1.4 Connected subsets.
3.1.5 Theorem (connected subsets of \(\br \)).
3.1.6 Separated subsets.
3.1.7 Proposition (connectedness and separated subsets).
3.1.8 Theorem (continuous preserve connectedness).
3.1.9 Corollary (Generalized Intermediate Value Theorem).
3.1.10 Theorem (union of connected sets).
3.1.11 Lemma (connectedness of closure).
3.1.12 Theorem (product of connected spaces).
3.1.13 Exercises.
3.2 Connected Components
3.2.1 Definition of components.
3.2.2 Proposition (properties of components).
3.2.3 Totally disconnected space.
3.2.4 Quasi-components.
3.2.5 Proposition (properties of quasi-components).
3.2.6 Exercises.
3.3 Path-connected Spaces
3.3.1 Paths.
3.3.2 Definition of path-connectivity.
3.3.3 Lemma.
3.3.4 Theorem (path-connected are connected)
3.3.5 Example (topologist’s sine curve).
3.3.6 Theorem (path-connectedness and continuity).
3.3.7 Theorem (path-connectedness and products).
3.3.8 Path components.
3.3.9 Theorem (Space-Filling curve).
3.3.10 Exercises.
3.4 Local Connectivity
3.4.1 Locally connected spaces.
3.4.2 Theorem (criterion for local connectedness).
3.4.3 Theorem (continuity and local connectedness).
3.4.4 Theorem (local connectedness and products).
3.4.5 Local path-connectedness.
3.4.6 Proposition (criterion for local path-connectedness).
3.4.7 Proposition (components of locally path-connected space)
3.4.8 Theorem (continuity and local path-connectedness).
3.4.9 Theorem (local path-connectedness and products).
3.4.10 Exercises.
4 Convergence
4.1 Sequences
4.1.1 Convergence of sequences.
4.1.2 Cluster points of sequences.
4.1.3 Proposition (sequences and closure in metric spaces).
4.1.4 Proposition (continuity and sequences in metric spaces).
4.1.5 Subsequences.
4.1.6 Proposition (subsequences and cluster points).
4.1.7 Exercises.
4.2 Nets
4.2.1 Directed set.
4.2.2 Definition of a net.
4.2.3 Convergence of nets in topological spaces.
4.2.4 Hausdorff spaces.
4.2.5 Theorem (uniqueness of limits in Hausdorff spaces).
4.2.6 Theorem (nets and closure).
4.2.7 Theorem (nets and continuity).
4.2.8 Theorem (convergence of nets in product spaces).
4.2.9 Cluster points of nets.
4.2.10 Subnet.
4.2.11 Theorem (cluster points and subnets).
4.2.12 Universal net (ultranet).
4.2.13 Exercises.
4.3 Filters
4.3.1 Definition of a filter.
4.3.2 Filter generated by a net.
4.3.3 Nbhd filter.
4.3.4 Comparing filters.
4.3.5 Convergence of filters.
4.3.6 Theorem.
4.3.7 Ultrafilters.
4.3.8 Theorem.
4.3.9 Theorem.
4.3.10 Zorn’s Lemma.
4.3.11 Theorem.
4.3.12 Theorem.
4.4 Hausdorff Spaces
4.4.1 Proposition.
4.4.2 Corollary.
4.4.3 Theorem (product of Hausdorff spaces).
4.4.4 \(T_{0}\) spaces and \(T_{1}\) spaces.
4.4.5 Proposition (characterization of \(T_{1}\) spaces).
5 Compactness
5.1 Compact Spaces
5.2 Countable Compact Spaces
5.3 Compact Metric Spaces
5.4 Locally Compact Spaces
5.5 Proper Maps