The question asks to identify the statement that declares "Every complete metric space is of second category". This is a fundamental result in topology and functional analysis.

Let's examine the statement itself and the options provided.

Understanding Complete Metric Spaces and Category

First, let's clarify the terms involved:

• Metric Space: A set \(X\) together with a distance function (metric) \(d\) that satisfies certain properties (non-negativity, identity of indiscernibles, symmetry, triangle inequality).
• Complete Metric Space: A metric space where every Cauchy sequence converges to a point within the space. Examples include the set of real numbers \(\mathbb{R}\) and Euclidean spaces \(\mathbb{R}^n\) with the standard metric.
• Nowhere Dense Set: A set \(A\) in a topological space \(X\) is nowhere dense if the interior of its closure is empty. Formally, \(\text{int}(\text{cl}(A)) = \emptyset\). Intuitively, a nowhere dense set is 'thin' everywhere; its closure contains no open sets.
• Set of First Category (Meager Set): A set that can be expressed as a countable union of nowhere dense sets.
• Set of Second Category: A set that is not of first category. This means it cannot be written as a countable union of nowhere dense sets. Spaces of second category are, in a sense, 'large' or 'non-meager'.

The statement "Every complete metric space is of second category" relates the property of completeness to the topological property of being of second category.

Analyzing the Options

Let's look at the given options:

• Baire category theorem: This is a key theorem in topology and analysis. There are several equivalent formulations, but a common one states that in a complete metric space (or a locally compact Hausdorff space), the union of a countable collection of nowhere dense sets has an empty interior. Another equivalent formulation is exactly the statement given in the question: Every complete metric space is of the second category.
• Cantor Theorem: There are several theorems named after Cantor. One famous one states that for any set \(A\), the cardinality of the power set of \(A\) is strictly greater than the cardinality of \(A\) (\(|\mathcal{P}(A)| > |A|\)). Another is the Cantor's intersection theorem for complete metric spaces, which deals with nested sequences of non-empty closed sets with diameters tending to zero. Neither of these relates to the concept of categories or the statement about complete metric spaces being of the second category.
• Fundamental theorem: This term is too general. There are fundamental theorems in many areas of mathematics (e.g., Fundamental Theorem of Calculus, Fundamental Theorem of Algebra). None of these are relevant to the category of topological spaces or complete metric spaces.
• Baire sandwich theorem: This is not a standard name for a theorem in topology or analysis. The "sandwich theorem" (or Squeeze Theorem) is typically associated with limits of sequences or functions in calculus. Baire is associated with the Baire Category Theorem.

Conclusion: Baire Category Theorem

Based on the definitions and standard mathematical theorems, the statement "Every complete metric space is of second category" is precisely one of the standard formulations of the Baire Category Theorem. The theorem highlights that complete metric spaces are topologically large; they cannot be "covered" by a countable collection of nowhere dense sets.

Therefore, the correct theorem is the Baire category theorem.

Revision Table: Key Concepts

Additional Information: Significance of Baire Category Theorem

The Baire Category Theorem is a powerful tool in functional analysis and topology. It has several important consequences:

• It implies that the real numbers \(\mathbb{R}\) are of the second category, meaning \(\mathbb{R}\) cannot be written as a countable union of nowhere dense sets. This shows that \(\mathbb{R}\) is "large" in a topological sense.
• It is used to prove other significant theorems, such as the Open Mapping Theorem and the Closed Graph Theorem in functional analysis (for Banach spaces, which are complete normed vector spaces).
• It shows that the set of rational numbers \(\mathbb{Q}\) in \(\mathbb{R}\) is of the first category (meager), while the set of irrational numbers \(\mathbb{R} \setminus \mathbb{Q}\) is of the second category (in fact, a dense \(G_\delta\) set, which is also second category). This illustrates that sets of second category can be topologically much larger than sets of first category, even if both are uncountable (like \(\mathbb{R}\) and \(\mathbb{R} \setminus \mathbb{Q}\)).