In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for **comparison of the topologies**.

A topology on a set may be defined as the collection of subsets which are considered to be "open". An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.

Let τ_{1} and τ_{2} be two topologies on a set *X* such that τ_{1} is contained in τ_{2}:

- .

That is, every element of τ_{1} is also an element of τ_{2}. Then the topology τ_{1} is said to be a **coarser** (**weaker** or **smaller**) **topology** than τ_{2}, and τ_{2} is said to be a **finer** (**stronger** or **larger**) **topology** than τ_{1}. ^{ [nb 1] }

If additionally

we say τ_{1} is **strictly coarser** than τ_{2} and τ_{2} is **strictly finer** than τ_{1}.^{ [1] }

The binary relation ⊆ defines a partial ordering relation on the set of all possible topologies on *X*.

The finest topology on *X* is the discrete topology; this topology makes all subsets open. The coarsest topology on *X* is the trivial topology; this topology only admits the empty set and the whole space as open sets.

In function spaces and spaces of measures there are often a number of possible topologies. See topologies on the set of operators on a Hilbert space for some intricate relationships.

All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.

Let τ_{1} and τ_{2} be two topologies on a set *X*. Then the following statements are equivalent:

- τ
_{1}⊆ τ_{2} - the identity map id
_{X}: (*X*, τ_{2}) → (*X*, τ_{1}) is a continuous map. - the identity map id
_{X}: (*X*, τ_{1}) → (*X*, τ_{2}) is an open map

Two immediate corollaries of this statement are

- A continuous map
*f*:*X*→*Y*remains continuous if the topology on*Y*becomes*coarser*or the topology on*X**finer*. - An open (resp. closed) map
*f*:*X*→*Y*remains open (resp. closed) if the topology on*Y*becomes*finer*or the topology on*X**coarser*.

One can also compare topologies using neighborhood bases. Let τ_{1} and τ_{2} be two topologies on a set *X* and let *B*_{i}(*x*) be a local base for the topology τ_{i} at *x* ∈ *X* for *i* = 1,2. Then τ_{1} ⊆ τ_{2} if and only if for all *x* ∈ *X*, each open set *U*_{1} in *B*_{1}(*x*) contains some open set *U*_{2} in *B*_{2}(*x*). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.

The set of all topologies on a set *X* together with the partial ordering relation ⊆ forms a complete lattice that is also closed under arbitrary intersections. That is, any collection of topologies on *X* have a *meet* (or infimum) and a *join* (or supremum). The meet of a collection of topologies is the intersection of those topologies. The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union.

Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.

- Initial topology, the coarsest topology on a set to make a family of mappings from that set continuous
- Final topology, the finest topology on a set to make a family of mappings into that set continuous

In topology and related areas of mathematics, a **product space** is the Cartesian product of a family of topological spaces equipped with a natural topology called the **product topology**. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

In topology and related branches of mathematics, a **topological space** may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general 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 topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, **weak topology** is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

In mathematics, a **base***B* of a topology on a set *X* is a collection of subsets of *X* that is stable by finite intersection. A base defines a topology on *X* that has, as open sets, all unions of elements of *B*.

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 topology and related areas of mathematics, the **quotient space** of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the **quotient topology**, that is, with the finest topology that makes continuous the canonical projection map. In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.

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 open; in Alexandrov topologies the finite restriction is dropped.

In topology and related areas of mathematics, a **subspace** of a topological space *X* is a subset *S* of *X* which is equipped with a topology induced from that of *X* called the **subspace topology**.

In general topology and related areas of mathematics, the **initial topology** on a set , with respect to a family of functions on , is the coarsest topology on *X* that makes those functions continuous.

In functional analysis and related areas of mathematics a **dual topology** is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space.

In functional analysis and related areas of mathematics, the **Mackey topology**, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology. A topological vector space (TVS) is called a **Mackey space** if its topology is the same as the Mackey topology.

In mathematics, particularly in functional analysis, a **bornological space** is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and functions, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces were first studied by Mackey. The name was coined by Bourbaki after *borné*, the French word for "bounded".

In general topology and related areas of mathematics, the **final topology** on a set , with respect to a family of functions into , is the finest topology on that makes those functions continuous.

In topology and related fields of mathematics, a **sequential space** is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology.

In mathematics, a **finite topological space** is a topological space for which the underlying point set is finite. That is, it is a topological space for which there are only finitely many points.

In mathematics, a linear map is a mapping *V* → *W* between two modules that preserves the operations of addition and scalar multiplication.

The **injective tensor product** of two topological vector spaces was introduced by Alexander Grothendieck and was used by him to define nuclear spaces.

In the field of functional analysis, a subfield of mathematics, a **dual system**, **dual pair**, or a **duality** over a field *K* is a triple consisting of two vector spaces over *K* and a bilinear map *b* : *X* × *Y* → *K* satisfying the following two separation axioms:

- (S1) if
*x*∈*X*is such that*b*(*x*,*y*) = 0 for all*y*∈*Y*then*x*= 0; - (S2) if
*y*∈*Y*is such that*b*(*x*,*y*) = 0 for all*x*∈*X*then*y*= 0.

In mathematics, specifically in order theory and functional analysis, the **order topology** of an ordered vector space is the finest locally convex topological vector space (TVS) topology on *X* for which every order interval is bounded, where an **order interval** in *X* is a set of the form [*a*, *b*] := { *z* ∈ *X* : *a* ≤ *z* and *z* ≤ *b* } where *a* and *b* belong to *X*.

- ↑ Munkres, James R. (2000).
*Topology*(2nd ed.). Saddle River, NJ: Prentice Hall. pp. 77–78. ISBN 0-13-181629-2.

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