Topology glossary : Boundary (topology) 

Part 1 -- topological concepts

• Continuous. A function from one space to another is continuous if the preimage of every open set is open.

• Homeomorphic. Two spaces X and Y are homeomorphic if there is a bijective map f : X -> Y such that f and f ^-1 are continuous. From the standpoint of topology, X and Y are the same. The function f is called a homeomorphism.

• Closure. The closure of a set is the intersection of all closed sets which contain it. It is the smallest closed set containing the original set.

• Interior. The interior of a set is the union of all open sets contained in it. It is the largest open set contained in the original set.

• Boundary. The boundary of a set is the set's closure minus its interior.

• Dense. A dense set is a set whose closure is the whole space.

• Nowhere dense. A nowhere dense set is a set whose closure has empty interior.

• Neighbourhood. A neighbourhood of a set S is a set containing an open set which in turn contains the set S. A neighbourhood of a point p is a neighbourhood of the 1-point set {p}.

• Punctured neighbourhood. A punctured neighbourhood of a point p is a neighbourhood of p, minus p. For instance, {x: 0<|x|<1} is a punctured neighbourhood of 0 in the real line, because {x : 0 ≤ |x| < 1} is a neighbourhood of 0.

• Sub-base. A set of open sets is a sub-base for a topology if every open set is a union of finite intersections of sets in the sub-base.

• Base, or Basis. A set of open sets is a base for a topology if every open set is a union of sets in the base.

• Local base. A set B of neighbourhoods of a point x of a topological space X is a local base at x if every neighbourhood of x contains some member of B.

• Locally finite. A collection of subsets of a space is locally finite if every point has a neighbourhood which meets only finitely many of the subsets.

• Cover. A collection {U_i} of sets is a cover (or covering), if their union is the whole space. An open cover is a cover {U_i} in which each U_i is an open set.

• Subcover. A cover K is a subcover of a cover L if every member of K is a member of L.

• Refinement. A cover K is a refinement of a cover L if every member of K is a subset of some member of L.

• Functionally separated. Two sets A and B in a space are functionally separated if there is a continuous function from the space into the interval [0,1] with the property that A is mapped to 0 and B is mapped to 1.

• Partition of unity. A partition of unity is a set of continuous functions from a space to [0,1] such that any point has a neighbourhood where all but a finite number are identically zero, and the sum of all them at every point is 1.

• Homotopic maps. Two continuous maps f, g : X -> Y are homotopic if there is a continuous map H : X × [0,1] -> Y, such that H(x,0) = f(x) and H(x,1) = g(x) for all x in X. The function H is called a homotopy between f and g.

Part 2 -- types of topological spaces

Topological spaces can be classified regarding the degree to which their points are separated, regarding their compactness, their overall size and their connectedness.

Separation axioms

For a detailed treatment, see Separation axiom. Some of these terms are defined differently in older mathematical literature; see History of the separation axioms[?].

• T₀. A space is T₀ if for every pair of distinct points in the space, there is an open set containing one but not the other.

• T₁. A space is T₁ if all its singletons are closed. T₁ spaces are always T₀.

• Hausdorff or T₂. A space is Hausdorff if every two distinct points have disjoint neighborhoods. Hausdorff spaces are always T₁.

• Regular. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods. Regular T₀ spaces are always Hausdorff.

• Tychonoff. A Hausdorff space is Tychonoff if whenever C is a closed set and p is a point not in C, then C and p are functionally separated. Tychonoff spaces are always regular.

• Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity. Normal T₁ spaces are always Tychonoff.

Compactness

• Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.

• Lindelöf. A space is Lindelöf if every open cover has a countable subcover.

• Compact. A space is compact if every open cover has a finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.

• Locally compact. A space is locally compact if every point has a local base consisting of compact neighborhoods. Locally compact Hausdorff spaces are always Tychonoff.

Size

• Separable. A space is separable if it has a countable dense subset.

• First-countable. A space is first-countable if every point has a countable local base.

• Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.

Connectedness

• Connected. A space X is connected if it isn't the union of a pair of disjoint nonempty open sets.

• Locally connected. A space is locally connected if every point has a local base consisting of connected sets.

• Totally disconnected. A space is totally disconnected if it has no connected subset with more than one point.

• Path-connected. A space X is path-connected if for every two points x,y in X, there is a path p from x to y, i.e., a continuous map p : [0,1] -> X with p(0) = x, and p(1) = y. Path-connected spaces are always connected.

• Locally path-connected. A space is locally path-connected if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.

• Simply-connected. A space X is simply connected if it is path-connected and every continuous map f : S¹ -> X is homotopic to a constant map.

• Contractible. A space X is contractible if the identity map on X is homotopic to a constant map. Contractible spaces are always simply connected.

Miscellaneous

• Metrizable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable.

• Locally metrizable. A space is locally metrizable if every point has a metrizable neighbourhood.

• Homogeneous. A space X is homogeneous if for every x and y in X there is a homeomorphism f : X -> X such that f(x) = y. Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous.

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