Topology
Heine–Borel Theorem
In ℝⁿ, a subset is compact if and only if it is closed and bounded
Bolzano–Weierstrass Theorem
Every bounded sequence in ℝⁿ has a convergent subsequence
Brouwer Fixed-Point Theorem
Every continuous map from a compact convex set to itself has a fixed point
Urysohn's Lemma
In a normal space, disjoint closed sets can be separated by a continuous function
Tychonoff's Theorem
An arbitrary product of compact spaces is compact in the product topology
Path-Connected Implies Connected
Every path-connected topological space is connected; continuous paths prevent disconnections
Connected Components
Every topological space partitions into maximal connected subsets, each of which is closed
Quotient Topology
The quotient topology makes the projection continuous and is universal for maps out of the quotient
Product Topology
A map into a product space is continuous iff every coordinate projection composed with it is continuous
Subspace Topology
The subspace topology on a subset S of X is the coarsest topology making the inclusion map continuous
Hausdorff Implies T1
Every Hausdorff (T2) space is a T1 space, meaning singletons are closed sets
Continuous Image of Compact is Compact
The continuous image of a compact set is compact, a cornerstone of analysis and topology
Compact Subset of Hausdorff is Closed
In a Hausdorff space every compact subset is closed, generalising the Heine–Borel direction compact implies closed
Baire Category Theorem
In a complete metric space, the intersection of countably many dense open sets is dense
Separable Spaces
A topological space is separable if it contains a countable dense subset
Second-Countable Spaces
A topological space is second-countable if its topology has a countable basis
Metrizable Spaces
A topological space is metrizable if its topology is induced by some metric
Normal Space
A topological space is normal if disjoint closed sets can be separated by disjoint open neighborhoods
Regular Space
A topological space is regular if a closed set and a disjoint point can be separated by open neighborhoods
Alexandrov One-Point Compactification
Adding a single point at infinity to a locally compact Hausdorff space yields a compact space
Continuous Bijection on Compact-Hausdorff is Homeo
A continuous bijection from a compact space to a Hausdorff space is automatically a homeomorphism