Intereting Posts

Asymptotic difference between a function and its binomial average
Cauchy's residue theorem with an infinite number of poles
Relation between two Riemannain connections
Showing every knot has a regular projection using differential topology
What is the length of a sine wave from $0$ to $2\pi$?
The formalism behind integration by substitution
Coupon collector problem for collecting set k times.
Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$
Find all the possible values of $(a,b,c,d)$.
How to integrate a three products
Riemman sum of $1/x^3$
If $n>m$, then the number of $m$-cycles in $S_n$ is given by $\frac{n(n-1)(n-2)\cdots(n-m+1)}{m}$.
Is Morera's theorem the inverse theorem of Goursat's theorem?
Is there a simple explanation why degree 5 polynomials (and up) are unsolvable?
Square root of a Mersenne number is irrational

Let $X$ be topological space and $Y$ be a subset of $X$ with $i\colon Y\to X$ the inclusion map. Show that the induced topology of $Y$ is characterized by the following property: A function $f\colon Z \to Y$ of a topological space $Z$ into $Y$ is continuous if and only if $i\circ f$ is continuous.

- A wedge sum of circles without the gluing point is not path connected
- Topology from fundamental system of neighbourhoods once and for all
- What do we mean when we say a differential form “descends to the quotient”?
- the equivalency of two definitions of locally closed sets
- Normal + Connected -> Uncountable
- What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?
- Are the complements of two homeomorphic compact, connected subsets of $\mathbb{R}^2$ homeomorphic?
- Pacman on a Mobius Strip
- $\mathbb R^2$ is not homeomorphic to $\mathbb R^3$.
- Topology and axiom of choice

It is also helpful to see this definition of induced topology on a subset as a special case of the notion of *initial topology* with respect to a set of partial functions $f_\lambda: X \to X_\lambda, \lambda \in \Lambda$, where $X$ is a set, and $X_\lambda, \lambda \in \Lambda$ is a family of topological spaces. This is the topology which has a subbase the sets $f^{-1}_\lambda(U)$ for all open $U$ in $X_\lambda$ and $\lambda \in \Lambda$. It has the universal property that a function $f: Y \to X$, where $Y$ is a topological space, is continuous if and only if $f_\lambda \circ f$ is continuous for all $\lambda$. This is how for example one defines the topology on a manifold when the $f_\lambda$ are a family of charts.

- Embed $S^{p} \times S^q$ in $S^d$?
- Intuition behind Descartes' Rule of Signs
- Proof: A convergent Sequence is bounded
- Is there a name for a group having a normal subgroup for every divisor of the order?
- $\ell^1$ vs. continuous dual of $\ell^{\infty}$ in ZF+AD
- A normal matrix with real eigenvalues is Hermitian
- Is the unit circle $S^1$ a retract of $\mathbb{R}^2$?
- A technical step in proving Hardy's inequality
- Holomorphic function has at most countably zeros
- Fourier cosine transforms of Schwartz functions and the Fejer-Riesz theorem
- product of harmonic functions
- $f:P(X)\to X$ property
- Generators of $\mathbb{Z}$ and $\mathbb{Z}$ when $\mathbb{Z}$, $\mathbb{Z}$ are f.g.
- Gaussian proof for the sum of squares?
- What are the coefficients of the polynomial inductively defined as $f_1=(x-2)^2\,\,\,;\,\,\,f_{n+1}=(f_n-2)^2$?