Topology Formula Cheat Sheet Formula Guide for Spaces

Topology formula cheat sheet explains open sets, compactness, continuity, and connectedness, helping learners understand the structure of spaces and properties that remain unchanged under transformations.

Topology Formula Cheat Sheet

Topic	Formula	Description
Topological Space	(X, τ)	Set with collection of open sets
Open Set	U ∈ τ	Contains neighborhood of each point
Closed Set	X − U	Complement of open set
Closure	cl(A)	Smallest closed set containing A
Interior	int(A)	Largest open set inside A
Boundary	∂A = cl(A) − int(A)	Edge points of set
Limit Point	Every neighborhood intersects A	Accumulation point
Dense Set	cl(A) = X	Everywhere dense
Nowhere Dense	int(cl(A)) = ∅	Sparse set
Basis	Collection generating topology	Open set builder
Subspace Topology	τY = {U∩Y}	Induced topology
Product Topology	τ = Πτᵢ	Product spaces
Quotient Topology	τ = {U ⊆ Y : f⁻¹(U) open}	Identifications
Continuous Function	f⁻¹(U) open	Preimage rule
Homeomorphism	Bijective + continuous + inverse continuous	Same topology
Topological Property	Invariant under homeomorphism	Structural property
Open Map	f(U) open	Image open
Closed Map	f(F) closed	Image closed
Embedding	Injective homeomorphism onto image	Structure preserving
First Countable	Countable neighborhood basis	Local property
Second Countable	Countable base	Global property
Separable Space	Countable dense subset	Density property
Lindelöf Space	Every cover has countable subcover	Compact-like
Compact Space	Every open cover has finite subcover	Key property
Heine-Borel	Closed & bounded ⇒ compact	ℝⁿ result
Sequential Compactness	Every sequence has convergent subsequence	Equivalent in ℝ
Connected Space	Cannot split into disjoint open sets	Single piece
Path Connected	Any two points connected by path	Strong connection
Components	Maximal connected subsets	Parts
Hausdorff Space	Points separable by neighborhoods	T₂ space
Regular Space	Point & closed set separable	T₃ space
Normal Space	Two closed sets separable	T₄ space
Urysohn Lemma	Separate sets via function	Construction
Tietze Extension	Extend continuous functions	Extension theorem
Tychonoff Theorem	Product of compact spaces compact	Major result
Metric Space	(X,d)	Distance defined
Open Ball	B(x,r) = {y : d(x,y)<r}	Neighborhood
Complete Space	Every Cauchy sequence converges	Completeness
Baire Category	Not union of nowhere dense sets	Important theorem
Fixed Point (Banach)	Contraction ⇒ unique point	Existence
Fundamental Group	π₁(X)	Loop structure
Simply Connected	No holes	π₁ = trivial
Covering Space	Local homeomorphism	Lift property
Homotopy	Continuous deformation	Same shape
Homology	Algebraic invariants	Holes count
Euler Characteristic	V − E + F	Topological invariant

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Conclusion Topology is the analysis of the properties that do not change after continuous transformations. It is concerned with such notions as continuity, compactness, and connectedness. Topology is frequently referred to as rubber-sheet geometry since it is possible to stretch shapes without rupture. It is extensively applied in mathematics and physics. Topology assists in the study of the spaces and their formations. It plays a role in contemporary studies and higher mathematics. Topological learning enhances the ability to think and reason in an abstract manner. It relates geometry to analysis. Data analysis and computer science make use of topology. It is also involved in physics and engineering. All in all, topology is a special and significant branch of mathematics. In summary, topology studies properties of space and continuity. It is important for advanced mathematics and modern scientific research.