Cone (topology)¶

In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8639
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Algebraic Topology → Mathematics

Core Idea¶

Cone (topology) is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point. In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by CX or by \operatorname{cone}(X) .

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Pinch-the-Tube Party Hat

Take a loop of string. Stretch it upward into a tube, then pinch the top of the tube together into one point. You get a party-hat shape. In topology, making a cone means doing that to any shape: stretch it into a tube, then squeeze one end down to a single point.

Stretch Then Squish to a Point

In topology, the math of shapes that can be stretched and bent, you can make a Cone from any shape. First you stretch the shape into a cylinder, like pulling a circle into a tube. Then you squeeze one end of the cylinder down to a single point, called the tip or vertex. Starting from a circle, you get something like a party hat. For a shape sitting in ordinary space, you can also picture it as drawing straight lines from every point of the shape to one extra point outside it.

Cylinder With One End Collapsed

In topology, especially algebraic topology, the cone of a space X — written CX or cone(X) — is made by stretching X into a cylinder and then collapsing one end to a single point. Formally, you take the cylinder X × [0,1] and glue the whole bottom face X × {0} to one point v, the vertex; everything on that face becomes the same point. For a circle, this gives the surface of an ordinary cone, like a party hat. If X is a nonempty compact subset of ordinary Euclidean space, the cone is the same (homeomorphic) as the union of straight line segments from each point of X to a fixed point outside X, as long as those segments only meet at that point. The construction works for any topological space, not just nice geometric ones.

In topology, and especially algebraic topology, the cone of a topological space X, denoted CX or cone(X), is intuitively obtained by stretching X into a cylinder and collapsing one end face to a point. Formally, it is the result of attaching the cylinder X × [0,1] along its face X × {0} to a single point v, the vertex, via the projection p: X × {0} → v; equivalently, it is the quotient of X × [0,1] in which X × {0} is identified to one point. The construction applies to any topological space, not only geometric ones. When X is a nonempty compact subspace of Euclidean space, the cone is homeomorphic to the union of line segments from points of X to any fixed point v not in X, provided these segments meet only at v. This concrete picture justifies the name, but the topological definition via the quotient of the cylinder is what identifies the construction in general.

Scope of Application¶

Properties. The cone is used in algebraic topology precisely because it embeds a space as a subspace of a contractible space.
Definitions. CX = (X \times [0,1])\cupp v = \varinjlim \bigl( (X \times [0,1]) \hookleftarrow (X\times {0}) \xrightarrow{p} v\bigr),.
Definitions. where v is a point (called the vertex of the cone) and p is the projection to that point.
Definitions. In other words, it is the result of attaching the cylinder X \times [0,1] by its face X\times{0} to a point v along the projection p: \bigl(.
Definitions. If X is a non-empty compact subspace of Euclidean space, the cone on X is homeomorphic to the union of segments from X to any fixed point v \not\in X.

Clarity¶

A clear use of Cone (topology) names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.

Manages Complexity¶

Cone (topology) compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—furthermore, every cone is contractible to the vertex point by the homotopy.—and the practical consequence—where v is a point (called the vertex of the cone) and p is the projection to that point. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit.

Abstract Reasoning¶

Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.
Check operation and conditions. In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to.

Knowledge Transfer¶

Within the home domain. Knowledge about Cone (topology) transfers literally when a new case preserves the same carrier type, relation, and recognition test. The cone is used in algebraic topology precisely because it embeds a space as a subspace of a contractible space. CX = (X \times [0,1])\cupp v = \varinjlim \bigl( (X \times [0,1]) \hookleftarrow (X\times {0}) \xrightarrow{p} v\bigr),. Beyond the home domain. No canonical parent is asserted for Cone (topology).

Relationships to Other Abstractions¶

Current abstraction Cone (topology) Domain-specific

Parents (1) — more general patterns this builds on

Cone (topology) is a kind of Topological Space Domain-specific

The cone on X is a topological space obtained from X times an interval by collapsing one end.

Hierarchy paths (5) — routes to 3 parentless roots

Cone (topology) → Topological Space → Closure

Show alternative paths (4)

Neighborhood in Abstraction Space¶

Cone (topology) sits in a moderately populated region (41^st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Topological & Functional-Analytic Spaces (13 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08