T1 Space Definitions Examples Generalisations To Other Kinds Of

A T₁ space is also called an accessible space or a Fréchet space and a R₀ space is also called an symmetric space. (The term Fréchet space also refers to an entirely different notion from functional analysis. For this reason, the term T₁ space is preferred).

1 Definitions

A topological space X is T₁ if and only if the following equivalent conditions are satisfied:

X is R₀ if and only if one of the following equivalent conditions is satisfied:

To state things slightly differentely: in any topological space we have, as properties of any two points, the following implications

If the first arrow can be reversed the space is R₀. If the second arrow can be reversed the space is T₀. If the composite arrow can be reversed the space is T₁. Clearly, a space is T₁ if and only if it's both R₀ and T₀.

2 Examples

The Zariski topologyIn mathematics, the Zariski topology is a structure basic to algebraic geometry, especially since 1950. It is named after its originator, Oscar Zariski. The Zariski topology is defined by defining the closed sets to be the sets consisting of the mutual ze on an algebraic varietyAn important and ancient subfield of mathematics is algebraic geometry. In classical algebraic geometry and perhaps also in modern algebraic geometry the main objects of study are algebraic varieties . An affine algebraic variety is defined to be an irred is T₁. To see this, note that a point with local coordinatesLocal coordinates are measurement indices into a local coordinate system or a local coordinate space''. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city (c₁,...,c_n) is the zero set of the polynomials x₁-c₁, ..., x_n-c_n. Thus, the point is closed. However, this example is well known as a space that is not Hausdorff (T₂).

For a more concrete example, let's look at the cofinite topology on an infinite set. Specifically, let X be the set of integers, and define the open sets O_A to be those subsets of X which contain all but a finite subset A of X. Then given distinct integers x and y:

the open set O_{x} contains y but not x, and the open set O_{y} contains x and not y;
equivalently, every singleton set {x} is the complement of the open set O_{x}, so it is a closed set;

so the resulting space is T₁ by each of the definitions above. This space is not T₂, because the intersection of any two open sets O_A and O_B is O_A∪B, which is never empty. Alternatively, the set of even integers is compact but not closed, which would be impossible in a Hausdorff space.

We can modify this example slightly to get an R₀ space that is neither T₁ nor R₁. Let X be the set of integers again, and using the definition of O_A from the previous example, define a basis of open sets G_x for any integer x to be G_x = O_{{x, x+1}} if x is an even number, and G_x = O_{{x-1, x}} if x is odd. Then the open sets of X are, unions of the basis sets

The resulting space is not T₀ (and hence not T₁), because the points x and x + 1 (for x even) are topologically indistinguishable; but otherwise it is essentially equivalent to the previous example.

3 Generalisations to other kinds of spaces

The terms "T₁", "R₀", and their synonyms can also be applied to such variations of topological spaces as uniform spaces, Cauchy spaces, and convergence space s. The characteristic that unites the concept in all of these examples is that limits of fixed ultrafilters (or constant nets) are unique (for T₁ spaces) or unique up to topological indistinguishability (for R₀ spaces).

As it turns out, uniform spaces, and more generally Cauchy spaces, are always R₀, so the T₁ condition in these cases reduces to the T₀ condition. But R₀ alone can be an interesting condition on other sorts of convergence spaces, such as pretopological space s.

General topology

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1 Definitions

2 Examples

3 Generalisations to other kinds of spaces

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