Weak Hausdorff space
In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed. In particular,
Hyperconnected space
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two proper closed sets (whether disjoint or non-disjo
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is o
First-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space is said to be first-countable if each point has
Pseudometric space
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In
Locally connected space
In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Dowker space
In the mathematical field of general topology, a Dowker space is a topological space that is T4 but not countably paracompact. They are named after Clifford Hugh Dowker. The non-trivial task of provid
Homological connectivity
In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups.
Volterra space
In mathematics, in the field of topology, a topological space is said to be a Volterra space if any finite intersection of dense Gδ subsets is dense. Every Baire space is Volterra, but the converse is
Feebly compact space
In mathematics, a topological space is feebly compact if every locally finite cover by nonempty open sets is finite. The concept was introduced by S. Mardeĉić and P. Papić in 1955. Some facts:
* Ever
Menger space
In mathematics, a Menger space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Menger space is a space in which for every sequence of open c
Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space (/ˈhaʊsdɔːrf/ HOWS-dorf, /ˈhaʊzdɔːrf/ HOWZ-dorf), separated space or T2 space is a topological space where for any two distinct point
Ultraconnected space
In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint. Equivalently, a space is ultraconnected if and only if the closures of two distinct points
Luzin space
In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 space without isolated points in which every nowhere-dense subset is countable. There are many m
Glossary of topology
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topolo
Dyadic space
In mathematics, a dyadic compactum is a Hausdorff topological space that is the continuous image of a product of discrete two-point spaces, and a dyadic space is a topological space with a compactific
Extremally disconnected space
In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word "extremally
Collectionwise Hausdorff space
In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete subset of , there is a pairwise disjoint family of open sets with each
Topological property
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological
Sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as
Paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact
D-space
In mathematics, a topological space is a D-space if for any family of open sets such that for all points , there is a closed discrete subset of the space such that .
Michael selection theorem
In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following: Let X be a paracompact
Totally disconnected space
In topology and related branches of mathematics, a totally disconnected space is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every
Separable space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the
Alexandrov topology
In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is op
Collectionwise normal space
In mathematics, a topological space is called collectionwise normal if for every discrete family Fi (i ∈ I) of closed subsets of there exists a pairwise disjoint family of open sets Ui (i ∈ I), such t
Σ-compact space
In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact and (weakly) locally
Locally normal space
In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that e
Baire space
In mathematics, a topological space is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.According to the Baire category theorem, compact Hausdor
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space is second-countable if
Supercompact space
In mathematics, in the field of topology, a topological space is called supercompact if there is a subbasis such that every open cover of the topological space from elements of the subbasis has a subc
Rothberger space
In mathematics, a Rothberger space is a topological space that satisfies a certain a basic selection principle. A Rothberger space is a space in which for every sequence of open covers of the space th
Pseudonormal space
In mathematics, in the field of topology, a topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing the
Sequentially compact space
In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in . Every metric space is naturally a topological spa
Paranormal space
In mathematics, in the realm of topology, a paranormal space is a topological space in which every countable discrete collection of closed sets has a locally finite open expansion.
Resolvable space
In topology, a topological space is said to be resolvable if it is expressible as the union of two disjoint dense subsets. For instance, the real numbers form a resolvable topological space because th
Fréchet–Urysohn space
In the field of topology, a Fréchet–Urysohn space is a topological space with the property that for every subset the closure of in is identical to the sequential closure of in Fréchet–Urysohn spaces a
Semi-locally simply connected
In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological
Polyadic space
In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.
Locally finite space
In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood, that is, an open neighborhood consisting of finitely many elements.
Shrinking space
In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open cover admits a shrinking. A shrinking of an open cover is another open cover indexed by the
Hemicompact space
In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set
Topological manifold
In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topologic
Toronto space
In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality. There are five homeomorphism classes o
Mesocompact space
In mathematics, in the field of general topology, a topological space is said to be mesocompact if every open cover has a compact-finite open refinement. That is, given any open cover, we can find an
End (topology)
In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically
H-closed space
In mathematics, a Hausdorff space is said to be H-closed, or Hausdorff closed, or absolutely closed if it is closed in every Hausdorff space containing it as a subspace. This property is a generalizat
Perfect set
In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set is perfect if , where denotes the set of all limit points of , also kn
Locally finite collection
In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension. A c
Countably compact space
In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
Zero-dimensional space
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimensi
Locally simply connected space
In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally co
Hurewicz space
In mathematics, a Hurewicz space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Hurewicz space is a space in which for every sequence of op
Uniformizable space
In mathematics, a topological space X is uniformizable if there exists a uniform structure on X that induces the topology of X. Equivalently, X is uniformizable if and only if it is homeomorphic to a
Compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space hav
P-space
In the mathematical field of topology, there are various notions of a P-space and of a p-space.
Kolmogorov space
In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them
Scattered space
In mathematics, a scattered space is a topological space X that contains no nonempty dense-in-itself subset. Equivalently, every nonempty subset A of X contains a point isolated in A. A subset of a to
Contractible space
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be con
Simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitive
Heine–Borel theorem
In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following two statements are equivalent:
* S is closed and bounde
Monotonically normal space
In mathematics, specifically in the field of topology, a monotonically normal space is a particular kind of normal space, defined in terms of a monotone normality operator. It satisfies some interesti
Simply connected at infinity
In topology, a branch of mathematics, a topological space X is said to be simply connected at infinity if for any compact subset C of X, there is a compact set D in X containing C so that the induced
Sober space
In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every irreducible closed subset has
Noetherian topological space
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open
Orthocompact space
In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open cover has an interior-preserving open refinement. That is, given an open cover of the top
Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which
Pseudocompact space
In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the spac
Locally Hausdorff space
In mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has an open neighbourhood that is a Hausdorff space under the subspace topology. Here are s
Homotopical connectivity
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has
Semiregular space
A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a base for the topology.
Selection principle
In mathematics, a selection principle is a rule assertingthe possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of selection pri
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one o
Rickart space
In mathematics, a Rickart space (after Charles Earl Rickart), also called a basically disconnected space, is a topological space in which open σ-compact subsets have compact open closures. named them
Door space
In mathematics, in the field of topology, a topological space is said to be a door space if every subset is open or closed (or both). The term comes from the introductory topology mnemonic that "a sub
Metacompact space
In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological sp
Regular space
In topology and related fields of mathematics, a topological space X is called a regular space if every closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods.
Ω-bounded space
In mathematics, an ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if P is some property of subspaces, then a P-bounded space is one i
Normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdor
Cellular space
A cellular space is a Hausdorff space that has the structure of a CW complex.
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Limit point compact
In mathematics, a topological space X is said to be limit point compact or weakly countably compact if every infinite subset of X has a limit point in X. This property generalizes a property of compac
Relatively compact subspace
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact.
Sub-Stonean space
In topology, a sub-Stonean space is a locally compact Hausdorff space such that any two open σ-compact disjoint subsets have disjoint compact closures. Related, an F-space, introduced by , is a comple
Realcompact space
In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regular Hausdorff and every point of its Stone–Čech compactification is real (meaning that t