Question

If \(X\) is a topological space, \(A \subset X\) is closed, and \(g \in C(A)\) satisfies \(g=0\) on \(\partial A\), then the extension of \(g\) to \(X\) defined by \(g(x)=0\) for \(x \in A^{c}\) is continuous.

Short answer

The extended function is continuous on the entire space \( X \).

Step-by-step solution

Understand the Problem

We want to show that a function defined on a closed subset of a topological space, and extended to the entire space by setting it to zero outside the subset, remains continuous. We have \( g \) defined on \( A \subset X \) and \( g \in C(A) \), meaning \( g \) is continuous on \( A \). Additionally, \( g = 0 \) on the boundary \( \partial A \) and we extend \( g \) by defining \( g(x) = 0 \) for \( x \in A^c \).

Analyze Continuity on A

Since \( g \) is continuous on \( A \), for every open set \( U \subseteq \mathbb{R} \) where \( g(A) \subseteq U \), the pre-image \( g^{-1}(U) \) is open in \( A \). Because \( A \) is closed in \( X \), any open set in the subspace topology of \( A \) corresponds to the intersection of an open set in \( X \) with \( A \).

Verify Boundary Condition

Given \( g = 0 \) on \( \partial A \), \( g \) satisfies the condition that \( g(x) = 0 \) for \( x \) on the boundary. This ensures the transition from \( A \) to \( A^c \) is smooth or continuous (since \( g = 0 \) on the boundary, it seamlessly maps to \( g(x) = 0 \) in \( A^c \)).

Check Continuity on A^c

On \( A^c \), the function is defined such that \( g(x) = 0 \). Since a constant function is continuous, \( g \) defined as zero on \( A^c \) is trivially continuous.

Conclude Overall Continuity

Combine the continuity from \( A \) and \( A^c \). Because \( g(x) = 0 \) on the boundary seamlessly transitions to \( A^c \), and \( g \) is continuous within \( A \), the entire extended function \( g \) is continuous on \( X \). The matching value of zero on the boundary keeps the transition between \( A \) and \( A^c \) continuous.

Key concepts

Continuity

When discussing topological spaces, one of the foundational concepts is continuity. A function is continuous if, intuitively, small changes in the input result in small changes in the output. More formally, a function \( f: X \rightarrow Y \) between topological spaces is continuous if for every open set \( V \subseteq Y \), the pre-image \( f^{-1}(V) \) is also an open set in \( X \).

In the context of our problem, continuity is preserved even after extending a function \( g \) from a closed subset to the entire space. For the function \( g \) to be continuous across \( X \), it must retain this property both on \( A \) and its complement \( A^c \) in \( X \). This is achieved by ensuring the values match smoothly at the boundary \( \partial A \), which we'll explore in other sections.

Ensuring that a function is continuous involves checking its behavior both on its domain and on extensions, requiring a synthesis of local and global perspectives on the function's behavior.

Closed Subset

A closed subset in a topological space has some very particular properties that make it distinct. A subset \( A \) of a topological space \( X \) is defined as closed if its complement \( A^c = X \setminus A \) is open. This is equivalent to saying that \( A \) contains all its limit points, meaning you cannot "narrow in" on points from outside \( A \) without first crossing into it.

In our exercise, the function \( g \) is defined on a closed subset \( A \). Since \( A \) is closed, this ensures that the boundary \( \partial A \) is well-defined and contained within \( A \). This attribute is key in ensuring the smooth extension of the function \( g \) to the whole space \( X \), as continuity conditions are straightforward to manage here.

• Closed sets support defining functions that align with the boundary conditions.
• The behavior of a function on closed sets directly influences its extendability.

Constant Function

A constant function is one of the simplest types of functions. In essence, a constant function takes any input and always returns the same single value. Mathematically, if a function \( f: X \rightarrow Y \) is constant, then there exists a \( c \in Y \) such that for every \( x \in X \), \( f(x) = c \).

The task describes extending function \( g \) over \( A^c \) as constant with value 0, i.e., \( g(x)=0 \) for all \( x \in A^c \). A constant function like this is trivially continuous because no matter how much the input changes, the output remains fixed. Thus, on \( A^c \), continuity is automatically satisfied.

• Constant functions provide simplicity and ease in ensuring continuity.
• They are essential in smoothly extending functions over larger domains with minimal complexity.

Boundary Condition

Boundary conditions play a critical role in understanding and solving many mathematical problems. They specify the behavior of a function along the boundary \( \partial A \) of its domain.

In our problem, the boundary condition specifies \( g(x) = 0 \) for every \( x \) on the boundary \( \partial A \). This allows for the seamless transition to \( g(x) = 0 \) on the outside of \( A \) (\( A^c \)). Essentially, these boundary conditions ensure that there is no discontinuity at the edges of \( A \).

• Boundary conditions are crucial for ensuring overall continuity when extending functions.
• They help to "glue" the function behavior across different parts of the space.
• Observing boundary conditions minimizes potential disruptions in transitions between regions.