Open set preimage

1 Definition and basic notation

1.1 Preimage of a set under a function

Given a function \(f:X\to Y\) and a subset \(A\subseteq Y\), the preimage (or inverse image) of \(A\) under \(f\) is the set of points in \(X\) that \(f\) sends into \(A\). It is written \[ f^{-1}(A)=\{x\in X: f(x)\in A\}. \] This construction converts questions about subsets of the codomain into corresponding subsets of the domain.

1.2 Open sets in topological spaces

A topological space is a pair \((X,\tau)\) where \(\tau\) is a collection of subsets of \(X\) designated as open sets, satisfying: \(\emptyset\) and \(X\) are open, arbitrary unions of open sets are open, and finite intersections of open sets are open. Similar structure is assumed for \((Y,\sigma)\).

1.3 Open set preimage \(f^{-1}(U)\)

Let \(U\subseteq Y\) be open. The open set preimage under \(f\) is simply the set \[ f^{-1}(U)=\{x\in X: f(x)\in U\}. \] Whether \(f^{-1}(U)\) is open in \(X\) depends on additional properties of \(f\) and is the key point in the topological definition of continuity.

1.4 Examples with standard topologies

• Identity map: If \(f=\mathrm{id}_X\), then for any open \(U\subseteq X\), \(f^{-1}(U)=U\), so the preimage of an open set is open.
• Inclusion map: If \(i:A\hookrightarrow X\) is an inclusion and \(U\subseteq X\) is open, then \(i^{-1}(U)=U\cap A\), which is open in \(A\) under the subspace topology.
• Constant map: If \(f(x)=y_0\) for all \(x\), then \(f^{-1}(U)\) is either \(X\) (if \(y_0\in U\)) or \(\emptyset\) (if \(y_0\notin U\)). This typically satisfies continuity but illustrates how preimages can collapse many points to one outcome.

2 Relationship to continuity

2.1 Continuity via open-set preimages

2.1.1 The open-set criterion for continuity

A map \(f:(X,\tau)\to (Y,\sigma)\) is continuous if and only if for every open set \(U\in\sigma\), the preimage \(f^{-1}(U)\) is open in \(X\). Symbolically: \[ \forall U\in\sigma,\quad f^{-1}(U)\in\tau. \] This criterion captures continuity purely in terms of how open sets in the codomain pull back to the domain.

2.1.2 Equivalent formulations using closed sets

Continuity can also be expressed using closed sets: \(f\) is continuous if and only if for every closed set \(C\subseteq Y\), the preimage \(f^{-1}(C)\) is closed in \(X\). This equivalence follows because complements exchange openness and closedness, and preimages commute with complements: \[ f^{-1}(Y\setminus C)=X\setminus f^{-1}(C). \]

2.2 Continuity of maps between metric spaces

2.2.1 Characterization using \(\varepsilon\)-\(\delta\) versus open sets

For metric spaces, continuity is often defined by the \(\varepsilon\)-\(\delta\) condition: for every \(y\) close to \(f(x_0)\), points \(x\) close to \(x_0\) map into that neighborhood. The open-set definition is equivalent: the metric topology defines open balls, and continuity is exactly the requirement that preimages of open balls are open. In practice, one may prove continuity either by showing that preimages of open sets are open, or by \(\varepsilon\)-\(\delta\) estimates.

2.3 Continuity and preimages of unions and intersections

2.3.1 Preimage distributes over unions

For any family of subsets \(\{A_i\}_{i\in I}\subseteq Y\), \[ f^{-1}\Big(\bigcup_{i\in I} A_i\Big)=\bigcup_{i\in I} f^{-1}(A_i). \] This identity is immediate from the definition: \(x\) maps into a union exactly when it maps into at least one member.

2.3.2 Preimage distributes over intersections

Similarly, for any family \(\{A_i\}_{i\in I}\subseteq Y\), \[ f^{-1}\Big(\bigcap_{i\in I} A_i\Big)=\bigcap_{i\in I} f^{-1}(A_i). \] These distribution rules are repeatedly used when checking openness or closedness of preimages.

3 Algebra of preimages

3.1 Preimage and set-theoretic operations

3.1.1 Preimage of complements

Preimages commute with complements: \[ f^{-1}(Y\setminus A)=X\setminus f^{-1}(A). \] Hence any statement about openness can be converted to a statement about closedness and vice versa.

3.1.2 Preimage of differences

For subsets \(A,B\subseteq Y\), \[ f^{-1}(A\setminus B)=f^{-1}(A)\setminus f^{-1}(B). \] This is useful for isolating points that land in one region of the codomain but not another.

3.2 Behavior under compositions of functions

3.2.1 \((g\circ f)^{-1}(U)=f^{-1}(g^{-1}(U))\)

If \(f:X\to Y\) and \(g:Y\to Z\), and \(U\subseteq Z\), then \[ (g\circ f)^{-1}(U)=f^{-1}(g^{-1}(U)). \] This identity allows continuity arguments to be built step-by-step: openness of preimages under \(g\circ f\) follows from openness under \(g\) and then under \(f\).

3.3 Order-preserving properties

3.3.1 Monotonicity with respect to set inclusion

Preimage is monotone: if \(A\subseteq B\subseteq Y\), then \[ f^{-1}(A)\subseteq f^{-1}(B). \] This ordering behavior underlies many “basis checking” arguments, where one proves the condition on a generating family of sets and extends it using set relations.

4 Open maps and their preimages

4.1 Definitions: open map vs continuous map

• A map \(f:X\to Y\) is continuous if preimages of open sets are open.
• A map \(f:X\to Y\) is an open map if images of open sets are open; that is, whenever \(V\subseteq X\) is open, \(f(V)\subseteq Y\) is open.

These notions are logically independent: a function can be continuous without being open, and it can be open without being continuous.

4.2 Preimage of open sets under open maps

4.2.1 When open maps preserve openness (and when they do not)

An open map does not automatically guarantee that \(f^{-1}(U)\) is open for open \(U\); that property is exactly continuity, not openness. However, open maps can interact with preimages through additional structure. For instance, if \(f\) is bijective and open, then it is also continuous (with continuous inverse), so preimages of open sets behave well. In general, without such extra hypotheses, the preimage of an open set may fail to be open even when the map sends open sets to open sets.

4.3 Examples and counterexamples

4.3.1 Non-injective versus injective effects on images

Non-injectivity can complicate images: an open set in \(X\) may map to a set in \(Y\) that is “folded” or “compressed” in a way that affects openness unless the topology in \(Y\) is compatible with the map. Injective maps can still fail to be open depending on how the map distorts neighborhoods relative to the target topology. These examples emphasize that openness concerns images, whereas continuity concerns preimages.

5 Special cases and common settings in analysis

5.1 Preimages under continuous functions on \(\mathbb{R}\)

5.1.1 Interval preimages and continuity

For functions \(f:\mathbb{R}\to\mathbb{R}\), open sets are unions of open intervals. Consequently, continuity is equivalent to the statement that for every open interval \((a,b)\), \[ f^{-1}((a,b)) \text{ is open in } \mathbb{R}. \] Often, one checks this using monotonicity or known algebraic forms (polynomials, exponentials, trigonometric functions), where the preimage of an interval can be described using inequalities.

5.2 Open sets generated by basis elements

5.2.1 Checking openness using a basis

If \(\mathcal{B}\) is a basis for the topology on \(Y\), then to verify continuity it suffices to check that for every basis element \(B\in\mathcal{B}\), the preimage \(f^{-1}(B)\) is open in \(X\). Since every open set is a union of basis elements, distribution of preimage over unions then yields openness for all open sets. This reduces the number of sets to check and is a common strategy in proofs.

5.3 Preimages of balls in metric spaces

5.3.1 \(f^{-1}(B_Y(y,r))\) and continuity criteria

In a metric space \((Y,d_Y)\), open balls \(B_Y(y,r)=\{z\in Y:d_Y(z,y)<r\}\) form a basis. For \(f:X\to Y\), continuity is equivalent to requiring that for every \(y\in Y\) and \(r>0\), \[ f^{-1}(B_Y(y,r)) \] is open in \(X\). This formulation aligns with the geometric intuition behind the metric \(\varepsilon\)-\(\delta\) definition.

6 Topological constructions involving open-set preimages

6.1 Subspace topology

6.1.1 Restriction of a function to a subspace

Let \(A\subseteq X\) with the subspace topology. If \(f:X\to Y\) is continuous, then the restriction \(f	_A:A\to Y\) is also continuous. For an open set \(U\subseteq Y\),

\[

(f	_A)^{-1}(U)=A\cap f^{-1}(U),

\] which is open in \(A\) because \(f^{-1}(U)\) is open in \(X\) and intersection with \(A\) produces a subspace-open set.

6.2 Quotient maps and saturation (basic relation)

6.2.1 Compatibility of preimages with quotient structures

Quotient spaces identify points in \(X\) according to an equivalence relation. A standard tool is the quotient map \(q:X\to X/{\sim}\). Open sets in the quotient are characterized so that their preimages under \(q\) are open in \(X\). In other words, \(W\) is open in the quotient if and only if \(q^{-1}(W)\) is open in \(X\). This “open-set via preimage” rule mirrors continuity criteria and shows how preimages govern the quotient topology.

6.3 Product spaces

6.3.1 Open sets in products and their preimages

For product spaces \(X\times Y\) with the product topology, open sets are built from products of open sets in the factors (more precisely, unions of sets of the form \(U\times V\)). When analyzing continuity or constructing continuous maps into or out of \(X\times Y\), one uses the fact that preimages interact well with products: checking preimages of basic open rectangles often suffices, then extends to arbitrary open sets by unions.

7 Related notions

7.1 Closed-set preimages and continuity variants

Since complements of open sets are closed sets, continuity using closed-set preimages is equivalent to continuity using open-set preimages. For many related classes of maps (such as embeddings or homeomorphisms), closed-set behavior and open-set behavior both matter, and preimage identities provide the bridge.

7.2 Semi-continuity and one-sided openness conditions

7.2.1 Preimages of open sets in semicontinuous settings

In semicontinuity, one replaces “preimage of every open set is open” with a one-sided condition involving either inequalities or preimages of certain types of sets (often involving upper or lower level sets). Although the exact formulation depends on conventions, the guiding idea remains that specific families of sets in the codomain must pull back to open (or closed) sets in the domain, in a restricted way.

7.3 Measurable functions (connection via \(\sigma\)-algebras)

7.3.1 Open sets and Borel measurability

Measurability is analogous in spirit to continuity: instead of requiring preimages of open sets to be open, one requires preimages of sets from a \(\sigma\)-algebra to lie in the domain’s \(\sigma\)-algebra. In particular, a function between topological spaces is Borel measurable if preimages of Borel sets (generated from open sets) are measurable. Because Borel \(\sigma\)-algebras are generated by open sets, understanding preimages of open sets is often the first step toward measurability arguments.

8 Worked examples and exercises

8.1 Computing \(f^{-1}(U)\) explicitly

Let \(f:\mathbb{R}\to\mathbb{R}\) be \(f(x)=x^2\) and let \(U=(1,4)\). Then \[ f^{-1}(U)=\{x\in\mathbb{R}:1<x^2<4\}. \] Solving inequalities gives \( -2<x<-1\) and \(1<x<2\), so \[ f^{-1}(U)=(-2,-1)\cup(1,2). \] This illustrates that computing preimages often reduces to solving constraints expressed in the codomain.

8.2 Verifying continuity using open-set preimages

Suppose \(f:\mathbb{R}\to\mathbb{R}\) is \(f(x)=x^3\). For an open interval \(U=(a,b)\), continuity can be checked by observing that \(f^{-1}(U)\) is described by \[ a<x^3<b. \] Because the cube function is strictly increasing and continuous, the solution set is an open interval of the form \((\sqrt[3]{a},\sqrt[3]{b})\), which is open. This provides a direct open-set preimage verification.

8.3 Using basis/preimage rules in problem solving

If \(Y\) has a basis \(\mathcal{B}\), then continuity can be proved by checking only \(f^{-1}(B)\) for \(B\in\mathcal{B}\). For example, in metric spaces it is enough to check preimages of open balls. In product spaces, checking preimages of rectangles \(U\times V\) where \(U,V\) are open in the factors often suffices, because general open sets are unions of such rectangles and preimages commute with unions.

8.4 Common pitfalls and subtle cases

• Confusing images with preimages: Openness of \(f(V)\) (image of an open set) is unrelated to openness of \(f^{-1}(U)\) (preimage of an open set) unless additional conditions hold.
• Assuming preimages preserve openness without continuity: A map can send open sets to open sets (be open) yet still fail the condition that preimages of open sets are open.
• Neglecting topology on the subspace/product/quotient: Preimages may be open in the larger space but need to be interpreted with respect to the induced or quotient topology in the smaller setting.