Intereting Posts

Tricky Gaussian; a good method?
Extending an automorphism to the integral closure
Why Lie algebras of type $B_2$ and $C_2$ are isomorphic?
Limit $c^n n!/n^n$ as $n$ goes to infinity
Meaning of variables and applications in lambda calculus
Ways to fill a $n\times n$ square with $1\times 1$ squares and $1\times 2$ rectangles
Differentiable function with nowhere differentiable derivitive
Modulus trick in programming
$p$ divides $ax+by+cz$
$^{\mathbb{N}}$ with respect to the box topology is not compact
Question about Conditional Expectation
On the equation $3x^3 + 4y^3 + 5z^3 = 0$
Prove existence of a real root.
If $f\tau$ is continuous for every path $\tau$ in $X$, is $f:X\rightarrow Y$ continuous?
Prove that $ \lim (s_n t_n) =0$ given $\vert t_n \vert \leq M $ and $ \lim (s_n) = 0$

I need to show that if $f: X \rightarrow Y$ is 1-1 and $X$ and $Y$ are metric spaces, then if $\forall A\subset X, f(\overline{A})=\overline{f(A)} $ then $f$ is homeomorphism.

1) Assume $f$ is 1-1 and $\forall A\subset X, f(\overline{A})=\overline{f(A)} $. I need to show that f is a homeomorphism.

It is sufficient to show that f is continuous and $f^{-1}$ as well is continuous. I cannot show these steps unless f is onto since I need to use $f(f^{-1}(A))=A $ often.

Please help me with this.

Thank you.

- If $C$ is a component of $Y$ and a component of $Z$, is it a component of $Y\cup Z$?
- Prove that two paths on opposing corners of the unit square must cross.
- Why $I = $ is a $1$-manifold and $I^2$ not?
- If a measure only assumes values 0 or 1, is it a Dirac's delta?
- Given a pair of continuous functions from a topological space to an ordered set, how to prove that this set is closed?
- If $S \times \Bbb{R}^k$ is homeomorphic to $T \times \Bbb{R}^k$ and $S$ is compact, can we conclude that $T$ is compact?

- Homeomorphism $\phi : T^2/A \to X/B$. What are $ T^2/A$ and $X/B$?
- $f:\bf S^1 \to \bf R$, there exist uncountably many pairs of distinct points $x$ and $y$ in $\bf S^1$ such that $f(x)=f(y)$? (NBHM-2010)
- Continuity of $f \cdot g$ and $f/g$ on standard topology.
- Countable compact spaces as ordinals
- Nice underestimated elementary topology problem
- every topological space can be realized as the quotient of some Hausdorff space.
- An open interval is an open set?
- Prove that $$ is not a compact subset of $\mathbb{R}$ with the lower limit topology, i.e. open sets are of the form $[a,b)$.
- What is a topology?
- Does a perfect compact metric space have a closed subset homeomorphic to a countable product of 2 point sets?

Suppose that $f:X\to Y$ is bijective. Then there are a few useful things to note/prove:

(1) $f$ is continuous if and only if $f\left(\overline{A}\right)\subseteq\overline{f(A)}$ for all $A\subseteq X$. In the same way, $f^{-1}:Y\to X$ is continuous if and only if $f^{-1}\left(\overline{A}\right)\subseteq\overline{f^{-1}(A)}$ for all $A\subseteq Y$.

(2) $f$ is an open map if and only if $f^{-1}:Y\to X$ is continuous.

(3) Given $A\subseteq X,B\subseteq Y$, we have $f(A)\subseteq B$ if and only if $A\subseteq f^{-1}(B)$.

Is that enough to get you started?

Now, let’s suppose that $f:X\to Y$ is 1-to-1, but not necessarily onto. Putting $Z=f(X)$, we have $f:X\to Z$ is a bijection, and $f^{-1}:Z\to X$ is perfectly well-defined. Now, I’m going to give slightly altered versions of the above hints (and add another one) to fit your change to the question. First, though, I’m going to define the following notation to avoid confusion:

Given a subset $S$ of a topological space $T$, we will now denote the

closure of $S$ in $T$by $\text{cl}_T(S)$. (Since we’re dealing with several different spaces, it’s a good idea to keep track of where things are.)

Now, we want to show that $f$ is a homeomorphism *from $X$ to $Z$* (rather than to $Y$) if and only if $$\forall A\subseteq X,\:f\bigl(\text{cl}_X(A)\bigr)=\text{cl}_Y\bigl(f(A)\bigr).\tag{#}$$

Now, let’s amend my list of useful notes/results-to-be-proved.

(1) $f:X\to Z$ is continuous if and only if $f\bigl(\text{cl}_X(A)\bigr)\subseteq\text{cl}_Z\bigl(f(A)\bigr)$ for all $A\subseteq X.$ In the same way, $f^{-1}:Z\to X$ is continuous if and only if $f^{-1}\bigl(\text{cl}_Z(A)\bigr)\subseteq\text{cl}_X\bigl(f(A)\bigr)$ for all $A\subseteq Z.$

(2) $f:X\to Z$ is an open map if and only if $f^{-1}:Z\to X$ is continuous.

(3) Given $A\subseteq X,B\subseteq Y$, we have $f(A)\subseteq B$ if and only if $A\subseteq f^{-1}(B)$. (This is also true if we replace $Y$ with $Z.$)

(4) $S\subseteq Z$ is closed in $Z$ if and only if there is some $F\subseteq Y$ such that $S=F\cap Z$ and $F$ is closed in $Y;$ thus, $\text{cl}_Z(S)=Z\cap\text{cl}_Y(S)$ for all $S\subseteq Z.$

Hopefully that will get you where you need to go.

- What is the recurrence relation in this problem?
- How to create an identity for $\sin \frac{x}{4}$
- Evaluating a logarithmic integral in terms of trilogarithms
- Finding the size of a Galois Group of a splitting field for a polynomial of degree 6.
- Approximation of conditional expectation
- Proof of: If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$.
- How to construct hyperbolically equidistant points on a line?
- Find the number of all subsets of $\{1, 2, \ldots,2015\}$ with $n$ elements such that the sum of the elements in the subset is divisible by 5
- Find trace of linear operator
- What is the purpose of free variables in first order logic?
- Alternative Proof of ${{p^\alpha-1}\choose{k}} \equiv ({-1})^k (mod \ p)$
- $x^p-x-1$ is irreducible over $\mathbb{Q}$
- If $F(x,y)=0$, prove $\frac{d^2y}{dx^2}=-\frac{F_{xx}F_y^2-2F_{xy}F_xF_y+F_{yy}F^2_x}{F_y^3}$
- Image of Jacobson Radical is the Jacobson Radical
- Higher dimensional cross product