This is a **list of general topology topics**, by Wikipedia page.

- Basic concepts
- Limits
- Topological properties
- Compactness and countability
- Connectedness
- Separation axioms
- Topological constructions
- Examples
- Uniform spaces
- Metric spaces
- Topology and order theory
- Descriptive set theory
- Dimension theory
- Combinatorial topology
- Foundations of algebraic topology
- Topology and algebra
- See also

- direct sum and the dual construction product
- subspace and the dual construction quotient
- Topological tensor product

- Discrete space
- Trivial topology
- Cofinite topology
- Finer topology
- Product topology
- Quotient space
- Unit interval
- Continuum (topology)
- Extended real number line
- Long line (topology)
- Sierpinski space
- Cantor set, Cantor space, Cantor cube
- Space-filling curve
- Topologist's sine curve
- Uniform norm
- Weak topology
- Strong topology
- Hilbert cube
- Lower limit topology
- Sorgenfrey plane
- Real tree
- Compact-open topology
- Zariski topology
- Kuratowski closure axioms
- Unicoherent
- Solenoid (mathematics)

- Topology glossary
- List of topology topics
- List of geometric topology topics
- List of algebraic topology topics
- Publications in topology

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In topology and related areas of mathematics, a **metrizable space** is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric such that the topology induced by is **Metrization theorems** are theorems that give sufficient conditions for a topological space to be metrizable.

In mathematics, **topology** is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

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 topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, **topological groups** are logically the combination of groups and topological spaces, i.e. they are group and topological spaces at the same time, s.t. the continuity condition for the group operations connect these two structures together and consequently they are not independent from each other.

In topology and related branches of mathematics, a **normal space** is a topological space *X* that satisfies **Axiom T _{4}**: every two disjoint closed sets of

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 Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.

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 precisely, it is a topological space in which every point has a compact neighborhood.

In mathematics, the **real line**, or **real number line** is the line whose points are the real numbers. That is, the real line is the set **R** of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.

The **Baire category theorem** (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

In mathematics, **Tychonoff's theorem** states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1937 paper of Eduard Čech.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In mathematics, a topological space is said to be a **Baire space**, if for any given countable collection of closed sets with empty interior in , their union also has empty interior in . Equivalently, a locally convex space which is not meagre in itself is called a Baire space. According to Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of a Baire space. Bourbaki coined the term "Baire 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 topological space, the singletons are connected; in a totally disconnected space, these are the *only* connected proper subsets.

In mathematics, a **Cantor space**, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a **Cantor space** if it is homeomorphic to the Cantor set. In set theory, the topological space 2^{ω} is called "the" Cantor space.

In the mathematical discipline of general topology, a **Polish space** is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

* Counterexamples in Topology* is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

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Images, videos and audio are available under their respective licenses.