The encyclopedia of mathematics claims, without proof, that an upper semicontinuous function on a completely regular topological space X is the pointwise limit of a decreasing sequence of continuous functions. I was able to find the proof (Bourbaki, General Topology, part II) only for the case when X is perfectly normal. Is the general statement above true, and if it is where can I find a proof?
2026-02-23 06:29:56.1771828196
Upper semicontinuous function as a poinwise limit of continuous fuctions
159 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 POINTWISE-CONVERGENCE
- I can't understand why this sequence of functions does not have more than one pointwise limit?
- Typewriter sequence does not converge pointwise.
- Fourier Series on $L^1\left(\left[0,1\right)\right)\cap C\left(\left[0,1\right)\right)$
- Analyze the Pointwise and Uniform Convergence of: $f_n(x) = \frac{\sin{nx}}{n^3}, x \in \mathbb{R}$
- Uniform Convergence of the Sequence of the function: $f_n(x) = \frac{1}{1+nx^2}, x\in \mathbb{R}$
- Elementary question on pointwise convergence and norm continuity
- Pointwise and Uniform Convergence. Showing unique limit.
- A sequence $f_k:\Omega\rightarrow \mathbb R$ such that $\int f_k=0 \quad \forall k\in \mathbb N $ and $\lim\limits_{k\to\infty} f_k \equiv1$.
- Show that partial sums of a function converge pointwise but not uniformly
- example of a sequence of uniformly continuous functions on a compact domain converging, not uniformly, to a uniformly continuous function
Related Questions in SEMICONTINUOUS-FUNCTIONS
- The reason for a certain requirement in upper-semicontinuity
- Lower semicontinuous submeasure is countably subadditive
- Prove that a function is upper semi-continuous
- How to remember which is lower/upper semicontinuity?
- Lower semicontinuity and partial minimization
- For a continuous function defined on [a,b] , is the set of points at which f(x)>d closed set?
- $f$ is LSC at $x$ if and only if $\lim_{\delta \to 0}\inf\{f(y) | y \in B(x,\delta)\}=f(x)$
- How to show lower semicontinuity: differentiability $\rightarrow$ continuity $\rightarrow$ lower semicontinuity?
- Subharmonic on $U$ iff subharmonic on each $U_{\alpha}$, where $(U_{\alpha})$ is an open cover of $U$
- One-Sided Notion of Topological Closure
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?
See below Problem 1.7.15.c from “General topology” by Ryszard Engelking (Heldermann Verlag, Berlin, 1989).