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General Topology

Field: Topology

The study of properties of spaces that are preserved under continuous deformations.

Sequence of Expressions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function. Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of neighborhoods: f is continuous at some point x ∈ X if and only if for any neighborhood V of f(x), there is a neighborhood U of x such that f(U) ⊆ V. Intuitively, continuity means no matter how "small" V becomes, there is always a U containing x that maps inside V. If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x) instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance. Note, however, that if the target space is Hausdorff, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous. In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, this is accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets. A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition. In detail, a function f: X → Y is sequentially continuous if whenever a sequence (x_{n}) in X converges to a limit x, the sequence (f(x_{n})) converges to f(x). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If X is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions. Instead of specifying the open subsets of a topological space, the topology can also be determined by a closure operator (denoted cl), which assigns to any subset A ⊆ X its closure, or an interior operator (denoted int), which assigns to any subset A of X its interior. In these terms, a function f ⁣:(X,cl)→(X′,cl′) f\colon (X,\mathrm {cl} )\to (X',\mathrm {cl} ')\, between topological spaces is continuous in the sense above if and only if for all subsets A of X f(cl(A))⊆cl′(f(A)).f(\mathrm {cl} (A))\subseteq \mathrm {cl} '(f(A)). That is to say, given any element x of X that is in the closure of any subset A, f(x) belongs to the closure of f(A). This is equivalent to the requirement that for all subsets A' of X' f−1(cl′(A′))⊇cl(f−1(A′)).f^{-1}(\mathrm {cl} '(A'))\supseteq \mathrm {cl} (f^{-1}(A')). Moreover, f ⁣:(X,int)→(X′,int′) f\colon (X,\mathrm {int} )\to (X',\mathrm {int} ')\, is continuous if and only if f−1(int′(A))⊆int(f−1(A))f^{-1}(\mathrm {int} '(A))\subseteq \mathrm {int} (f^{-1}(A)) for any subset A of X. - ^Moore, E. H.; Smith, H. L. (1922). "A General Theory of Limits". American Journal of Mathematics. 44 (2): 102–121. doi:10.2307/2370388. JSTOR2370388. - ^Heine, E. (1872). "Die Elemente der Functionenlehre."Journal für die reine und angewandte Mathematik. 74: 172–188.
A set VV is a neighborhood of xx if there is an open set UU such that x∈U⊆Vx \in U \subseteq V.
Property

Properties

Let (X,τ)(X, \tau) be a topological space. A topological property PP is a condition that can be formulated as a statement about the open sets τ\tau or the underlying set XX. Formally, we define the set of all topological properties Prop\text{Prop} as the collection of predicates P:Top→BoolP: \text{Top} \to \text{Bool} such that P(X,τ)P(X, \tau) is true if and only if the space (X,τ)(X, \tau) satisfies the condition. For instance, the property of compactness, denoted Comp\text{Comp}, is defined by the condition that every open cover {U}i∈I\{U\}_{i \in I} of XX has a finite subcover. This can be expressed axiomatically as:\n\nComp(X,τ)≜∀U⊆τ,((⋃U∈UU=X)  ⟹  ∃F⊆U such that F is finite and ⋃U∈FU=X)\text{Comp}(X, \tau) \triangleq \forall \mathcal{U} \subseteq \tau, \left( \left(\bigcup_{U \in \mathcal{U}} U = X \right) \implies \exists \mathcal{F} \subseteq \mathcal{U} \text{ such that } \mathcal{F} \text{ is finite and } \bigcup_{U \in \mathcal{F}} U = X \right) \n\nSimilarly, the property of Hausdorff separation (T2_2) requires that for any two distinct points x,y∈Xx, y \in X, there exist disjoint open neighborhoods UU and VV such that x∈Ux \in U and y∈Vy \in V. This is formalized as:\n\nT2(X,τ)≜∀x,y∈X,(x≠y  ⟹  ∃U,V∈τ such that x∈U,y∈V, and U∩V=∅)\text{T}_2(X, \tau) \triangleq \forall x, y \in X, \left( x \neq y \implies \exists U, V \in \tau \text{ such that } x \in U, y \in V, \text{ and } U \cap V = \emptyset \right) \n\nThus, the set of properties Prop\text{Prop} is the set of all such predicates that characterize specific topological structures or constraints on the open sets τ\tau.
e.g., T2 (Hausdorff): Any two distinct points have disjoint neighborhoods.
Let (X,τ)(X, \tau) be a topological space. \n\n1. **First-Countability Axiom:** XX is first-countable if for every point x∈Xx \in X, there exists a countable family of open sets textBx={Un}n∈N\\text{B}_x = \{U_n\}_{n \in \mathbb{N}} (a local basis at xx) such that for every open neighborhood VV of xx, there exists an n∈Nn \in \mathbb{N} such that Un⊆VU_n \subseteq V. \n\n2. **Second-Countability Axiom:** XX is second-countable if there exists a countable basis textB={Vn}n∈N\\text{B} = \{V_n\}_{n \in \mathbb{N}} for the topology τ\tau. This means that for every open set W∈τW \in \tau, WW can be written as a union of elements from textB\\text{B}: W=⋃n∈NVn′W = \bigcup_{n \in \mathbb{N}} V_n' where textB′={Vn′}n∈N⊆B\\text{B}' = \{V_n'\}_{n \in \mathbb{N}} \subseteq \text{B} and Vn′∈BV_n' \in \text{B}.\n\nThese axioms characterize spaces whose topological structure is sufficiently 'small' or 'well-behaved' relative to countable sets, often implying properties like separability or metrizability under additional conditions.
In a complete metric space, the intersection of countably many dense open sets is dense.