General topology Basic notions: open sets, neighbourhoods, bases Concept
Topological space
Summary
A topological space is a set together with a choice of subsets, called open sets, that obey simple rules for unions and intersections. This structure is enough to define continuity, convergence, compactness and connectedness without a notion of distance.
Where this leads
167 pages build on this one.
- Completely normal space
- Homotopy lifting for covering maps
- Relative CW complex
- Regular open algebra
- Topological dynamics
- Locally path-connected space
- Vietoris topology
- Homotopy
- Germ of a holomorphic function
- Suspension
- Namioka--Phelps
- Natural duality
- Relative homotopy group
- Perfect map
- Moore space
- Compactification
- Urysohn space
- Filter limit
- Brouwer's characterization of the Cantor space
- Ringed space
- Kolmogorov quotient
- Whitney C-infinity topology
- Fiber bundle
- Perfect set
- Limit point compact space
- Hyperconnected space
- Stable homotopy theory
- Locale
- Adjunction space
- Generic point
- Frame
- Semiconjugacy and factor map
- Stone space
- Topological monoid
- Hurewicz fibration
- Continuous multiplication
- Tychonoff space
- Künneth theorem
- Local homeomorphism
- Sheaf
- Specialization preorder
- Homotopy lifting property
- Regular open set
- G-delta set
- Clopen set
- Fixed-point property
- Final topology
- Lebesgue covering dimension
- Topological vector space
- Scott topology
- Convergence of filters
- Serre fibration
- Topological semigroup
- Germ of an analytic set
- Linear topology
- Disjoint union topology
- Kantorovich duality
- Sigma-compact space
- Monad in nonstandard topology
- Topological homogeneity
- Non-wandering set
- Borel measure
- Semitopological group
- Varadhan's lemma
- Heyting algebra
- Hochster's theorem on spectral spaces
- Topological conjugacy
- Brenier's theorem
- Alexandrov topology
- Hemicontinuity
- Extremally disconnected space
- Topological transitivity
- Space of bounded continuous functions
- Locally connected space
- Topology of pointwise convergence
- Standard Borel space
- Compactness via ultrafilters
- Presheaf
- Countably compact space
- Perfectly normal space
- Priestley space
- Order topology
- Homotopy group with coefficients
- Manifold with corners
- Irreducible analytic germ
- Lindelöf space
- Local coefficient system
- Zariski topology
- Stalk
- Fine topology
- Thom spectrum
- Kuratowski closure axioms
- Net
- Little cubes operad
- Sequential space
- Initial topology
- Universal coefficient theorem
- Compactly generated space
- Urysohn metrization theorem
- Minimal set
- Fréchet–Urysohn space
- n-connected space
- Metacompact space
- Derived set
- Local trivialization
- Category of topological spaces
- Optimal transport and the Monge problem
- Pseudogroup
- Nagata–Smirnov metrization theorem
- Partition of unity
- Topological transformation group
- Locally finite family
- Noetherian topological space
- Wandering set
- I-adic topology
- Stone–Čech compactification
- Large deviations theory
- Filter
- Sober space
- Alexander subbase theorem
- Topology generated by a family of seminorms
- Topological mixing
- Topological group
- Residual set
- Displacement convexity
- Paratopological group
- Atlas
- Pointed space
- Kuratowski's closure-complement problem
- Cardinal function
- Haefliger structure
- Triangulation
- Lawson topology
- Sanov's theorem
- Spectral space
- Bing metrization theorem
- Baire property
- Cramér's theorem
- Relaxation of a variational problem
- Toda bracket
- Residuated lattice
- Inductive dimension
- Alexander–Spanier cohomology
- Nerve theorem
- Semilocally simply connected space
- Proper map
- Supermanifold
- Čech cohomology
- Compact-open topology
- Sheaf cohomology in topology
- Polish space
- Zero-dimensional space
- Tube lemma
- Rate function
- Pseudocompact space
- Convergence along an ultrafilter
- Mayer–Vietoris sequence
- F-sigma set
- Geometric realization
- Germ
- Manifold with boundary
- Semicontinuity
- Spectrum
- Topological manifold
- Support of a measure
- Paracompact space
- Irreducible component
Further reading
- Web https://en.wikipedia.org/wiki/Topological_space, lead section
About this page
| Tag | 0CQHFPGJ45 |
|---|---|
| URI | m2://0CQHFPGJ45 |
| Aliases | topology |
| Kind | concept |
| Writing | W1 stated |
| MSC | 54A05 |
| Same as |
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What this page needs for W2
- edge not_instance_of in: at least one non-example (0 of 1)
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What links here
Special cases
- Regular space
- T0 space
- Metrizable space
- First-countable space
- Locally compact space
- Normal space
- Totally disconnected space
- Compact space
- Connected space
- Separable space
- Baire space
- Initial topology
- Final topology
- Homology manifold
- Manifold with boundary
- Zariski topology
- Alexandrov topology
- Vietoris topology
- Limit point compact space
- Spectral space
- Kolmogorov quotient
- Adjunction space
- Suspension
- Topological manifold
- Order topology
- Compact-open topology
- Pointed space
- Sequential space
- Lawson topology
- Sigma-compact space
Instances
- Torus
- Möbius strip
- Cofinite topology
- Discrete topology
- Circle as a quotient of the real line
- Indiscrete topology
- Particular point topology
- Excluded point topology
- Sierpiński space
Definitions
Used by
- Completely normal space
- Homotopy lifting for covering maps
- Regular open algebra
- Topological dynamics
- Locally path-connected space
- Vietoris topology
- Homotopy
- Suspension
- Natural duality
- Relative homotopy group
- Perfect map
- Moore space
- Compactification
- Urysohn space
- Filter limit
- Brouwer's characterization of the Cantor space
- Ringed space
- Kolmogorov quotient
- Whitney C-infinity topology
- Fiber bundle
- Perfect set
- Limit point compact space
- Hyperconnected space
- Locale
- Adjunction space
- Generic point
- Frame
- Semiconjugacy and factor map
- Stone space
- Topological monoid
- Hurewicz fibration
- Continuous multiplication
- Tychonoff space
- Künneth theorem
- Local homeomorphism
- Sheaf
- Specialization preorder
- Homotopy lifting property
- Regular open set
- G-delta set
- Clopen set
- Fixed-point property
- Final topology
- Lebesgue covering dimension
- Topological vector space
- Scott topology
- Convergence of filters
- Serre fibration
- Topological semigroup
- Linear topology
- Disjoint union topology
- Sigma-compact space
- Monad in nonstandard topology
- Topological homogeneity
- Non-wandering set
- Borel measure
- Semitopological group
- Varadhan's lemma
- Heyting algebra
- Hochster's theorem on spectral spaces
- Topological conjugacy
- Alexandrov topology
- Hemicontinuity
- Extremally disconnected space
- Topological transitivity
- Space of bounded continuous functions
- Locally connected space
- Topology of pointwise convergence
- Standard Borel space
- Compactness via ultrafilters
- Presheaf
- Countably compact space
- Perfectly normal space
- Priestley space
- Order topology
- Manifold with corners
- Lindelöf space
- Local coefficient system
- Zariski topology
- Stalk
- Fine topology
- Kuratowski closure axioms
- Net
- Little cubes operad
- Sequential space
- Initial topology
- Universal coefficient theorem
- Compactly generated space
- Urysohn metrization theorem
- Minimal set
- Fréchet–Urysohn space
- n-connected space
- Metacompact space
- Derived set
- Local trivialization
- Category of topological spaces
- Pseudogroup
- Nagata–Smirnov metrization theorem
- Partition of unity
- Topological transformation group
- Locally finite family
- Noetherian topological space
- Wandering set
- I-adic topology
- Stone–Čech compactification
- Large deviations theory
- Filter
- Sober space
- Alexander subbase theorem
- Topology generated by a family of seminorms
- Topological mixing
- Topological group
- Residual set
- Paratopological group
- Atlas
- Pointed space
- Kuratowski's closure-complement problem
- Cardinal function
- Haefliger structure
- Triangulation
- Lawson topology
- Spectral space
- Bing metrization theorem
- Baire property
- Relaxation of a variational problem
- Inductive dimension
- Alexander–Spanier cohomology
- Semilocally simply connected space
- Proper map
- Supermanifold
- Čech cohomology
- Compact-open topology
- Sheaf cohomology in topology
- Polish space
- Zero-dimensional space
- Tube lemma
- Rate function
- Pseudocompact space
- Convergence along an ultrafilter
- Mayer–Vietoris sequence
- F-sigma set
- Germ
- Manifold with boundary
- Semicontinuity
- Topological manifold
- Support of a measure
- Paracompact space
- Irreducible component