I have been seeing this fact used a lot but have not been able to find a proper proof justifying it. Would anyone be able to outline one?
2026-02-23 08:25:16.1771835116
A $G_{\delta}$ subset of a Baire space is Baire
366 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GENERAL-TOPOLOGY
- Is every non-locally compact metric space totally disconnected?
- Let X be a topological space and let A be a subset of X
- Continuity, preimage of an open set of $\mathbb R^2$
- Question on minimizing the infimum distance of a point from a non compact set
- Is hedgehog of countable spininess separable space?
- Nonclosed set in $ \mathbb{R}^2 $
- I cannot understand that $\mathfrak{O} := \{\{\}, \{1\}, \{1, 2\}, \{3\}, \{1, 3\}, \{1, 2, 3\}\}$ is a topology on the set $\{1, 2, 3\}$.
- If for every continuous function $\phi$, the function $\phi \circ f$ is continuous, then $f$ is continuous.
- Defining a homotopy on an annulus
- Triangle inequality for metric space where the metric is angles between vectors
Related Questions in BAIRE-CATEGORY
- Is there an uncountable collection of pairwise disjoint second category subsets of Cantor space?
- What is the Baire characteristic of the real number line?
- Uniqueness in Baire property representation for compact Hausdorff spaces
- On the notion of Cech-complete
- Applications of Baire Category Theorem to Connected Sets
- If $f \in C^\infty (\mathbb{R})$ such that $f^{(n)}=0$ for some $n$, then $f$ is a polynomial?
- A question on the proof of the Baire Category theorem
- Proof of Uniform boundedness principle (why are the sets closed)
- Measure of complement of union of nowhere dense set with positive measure
- A Cech-complete subspace is a $G_\delta$ in its closure
Related Questions in THE-BAIRE-SPACE
- A Homeomorphism Between the Baire Space and a Countable $G_\delta$ set of the Cantor Space
- Gale-Stewart game
- Baire Space and Its Equivalent Property
- The Baire space as an automorphism group?
- Two Baire measure, $\left\{A\subseteq X: A \text{ Baire set}\ \mu(A)=\nu(A)\right\}$ is $\sigma$-algebra
- $S_\infty$ is a non-locally compact Polish group (Kechris)
- Baire space $\mathbb{N}^\mathbb{N}$ written as $\mathbb{R}$
- What can I say about $A^{c}$ if $A$ is not open
- $H$ thin in $X$ implies that $X\setminus H$ contains a generic subset
- Let $X$ be a complete metric space. $E$ is a non empty open set in $X$ Then is $E$ a first category set or second category set.
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
The complement of a countable intersection of dense open sets (i.e a $G_{\delta}$ set ) is a countable union of closed sets with empty interior, i.e. of the first Baire category. Therefore a $G_{\delta}$ subset of a Baire space is of the second Baire category, i.e. itself a Baire space.