Topological Space

A topological space is a set of points equipped with a structure called a topology, which specifies which subsets are open. This structure makes it possible to define continuity and convergence without assigning numerical distances between points. Topological spaces generalize metric spaces: every metric determines a topology, but some topologies cannot arise from any metric. The distinction allows spatial relationships to be studied independently of length, angle, or other quantitative measurements. (math.ucla.edu)

Definition and axioms

Formally, a topological space is a pair $(X,\tau)$, where $X$ is a set and $\tau$ is a collection of subsets of $X$. In the terminology of set theory, $\tau$ is a subset of the power set $\mathcal P(X)$. Its members are called open sets, and they must satisfy three axioms:

1. Both $\varnothing$ and $X$ belong to $\tau$.
2. The union of any collection of members of $\tau$ belongs to $\tau$.
3. The intersection of any finite collection of members of $\tau$ belongs to $\tau$.

The set $X$ is the underlying set, while $\tau$ is its topology. The same underlying set can carry different topologies and therefore represent different topological spaces. (pi.math.cornell.edu)

A closed set is a subset whose complement in $X$ is open. Consequently, arbitrary intersections and finite unions of closed sets are closed. “Open” and “closed” are not mutually exclusive classifications: a subset can be both, or neither. In particular, $X$ and $\varnothing$ are always both open and closed. (pi.math.cornell.edu)

Examples and bases

The usual topology on the real numbers consists of all unions of open intervals. More generally, a metric $d$ defines a topology by declaring $U$ open when each $x\in U$ has an open ball $B_d(x,r)$ contained in $U$, for some $r>0$. A space whose topology is induced by a metric is called metrizable. Different metrics can induce exactly the same topology. (math.ucla.edu)

Two extreme examples exist on every set. The discrete topology declares every subset open; the indiscrete topology has only $\varnothing$ and $X$ as open sets. On a set with at least two points, these topologies are distinct. Another example on $X=\{0,1\}$ is

which satisfies the axioms but treats its two points asymmetrically. (pi.math.cornell.edu)

A basis is a collection of open sets such that every open set is a union of members of the collection. Open intervals form a basis for the usual real-line topology, and open balls form a basis for a metric topology. Bases describe a topology economically: one need not list every open subset separately. (math.ucla.edu)

Neighborhoods, closure, and convergence

A neighborhood of a point $x$ is a set containing an open set that contains $x$; a neighborhood need not itself be open. The interior of a subset $A$ is the largest open set contained in $A$. Its closure, written $\overline A$, is the smallest closed set containing $A$. Equivalently, $x\in\overline A$ precisely when every open neighborhood of $x$ intersects $A$. A dense subset has closure equal to the whole space. (math.ucla.edu)

A sequence $(x_n)$ converges to $x$ if every neighborhood of $x$ contains all sufficiently late terms. This extends the familiar notion of a limit, but sequences do not capture every topological phenomenon in arbitrary spaces. First-countable spaces, in which each point has a countable local neighborhood basis, permit closure and continuity to be characterized using sequences; general spaces may require more general convergence devices. (math.ucla.edu)

Continuity and equivalence

A function $f:X\to Y$ between topological spaces is a continuous function if

For metric spaces, this agrees with the usual epsilon–delta definition. The definition uses inverse images: a continuous function need not send open sets to open sets. (math.ucla.edu)

A homeomorphism is a continuous bijection whose inverse is also continuous. Homeomorphic spaces are equivalent as topological spaces, even when their geometric measurements differ. A continuous bijection alone is insufficient; however, a continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism. (math.ucla.edu)

Constructing new spaces

Several constructions equip related sets with natural topologies:

• Subspace topology: For $A\subseteq X$, the open subsets of $A$ are the intersections $A\cap U$, with $U$ open in $X$. Openness is therefore relative to the ambient space.
• Product topology: A Cartesian product $\prod_i X_i$ receives the coarsest topology making all coordinate projections continuous. Basic open sets restrict only finitely many coordinates.
• Quotient topology: An equivalence relation on $X$ produces a set $X/{\sim}$. A subset $V$ of this quotient is open exactly when its inverse image under the canonical projection is open in $X$.

These constructions respectively describe taking subsets, combining spaces, and identifying points. (math.mit.edu)

Separation, compactness, and connectedness

A Hausdorff space is one in which any two distinct points have disjoint open neighborhoods. Every metric space is Hausdorff. In Hausdorff spaces, convergent sequences have unique limits, and compact subsets are closed. These conclusions need not hold in arbitrary topological spaces. (math.mit.edu)

A compact space is one for which every cover by open sets has a finite subcover. Compactness is not defined by boundedness: the equivalence between compactness and being closed and bounded applies to subsets of finite-dimensional Euclidean space, not to all topological spaces. (math.ucla.edu)

A connected space cannot be expressed as the union of two disjoint, nonempty open subsets. Path-connectedness requires any two points to be joined by a continuous path and is stronger than connectedness. Continuous images preserve both compactness and connectedness, making these properties useful for distinguishing spaces that cannot be homeomorphic. (math.ucla.edu)