# Tagged Questions

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

5 views

### Proving a set is connected using the definition of Relatively Open set Of a set in $C$

I came across this definition Let $U\subset S\subset C$. We say that $U$ is relatively open in $S$ if for every $z_0 \in U$, there is $r > 0$ such that $$D(z_0 ;r)\cap S\subset U$$: Now the ...
15 views

12 views

### Twist of irreducibility in compactifications

I am looking for a connected metric space $X$ that is (1) irreducible between two of its point $p$ and $q$ (meaning no proper closed connected subset of $X$ contains $p$ and $q$), such that (2) $X$ ...
13 views

### Lower limit topology with completely normal

I am trying to prove that lower limit topology is completely normal. I know the it is normal. I attempted to consider this as cases let $X$ completely normal and $Y$ subset of $X$ If $Y$ countable ...
21 views

### A question about retraction mapping

Let $f:\overline{B}_r(0)\rightarrow \overline{B}_r(0)\setminus\{0\}$ be a continuous function. Suppose that $f(x)=x$, when $x\in \partial \overline{B}_r(0)$. Then we can consider a continuous ...
16 views

22 views

### When proximal continuity and (topological) continuity are the same?

Under which conditions proximal continuity of $f$ (having $X\mathrel{\delta_1}Y \Rightarrow f[X]\mathrel{\delta_2}f[Y]$ for every sets $X$, $Y$ on the first proximity) from a proximity $\delta_1$ to a ...
33 views

### Baire Category theorem and open vs. closed nowhere dense sets

In Folland's book, part (b) of the Baire Category theorem states that $X$ is not a countable union of nowhere dense sets. where $X$ is a complete metric space. It doesn't say whether those sets ...
19 views

### Example for hollow sets whose complement is not dense in $\mathbb{R}$.

A set is hollow if it has empty interior. A set is no where dense (closure is hollow) if and only if i̶t̶s̶ ̶c̶o̶m̶p̶l̶e̶m̶e̶n̶t̶ ̶i̶s̶ ̶d̶e̶n̶s̶e̶ .̶ its complement contains a dense open set. ...
37 views

### An uncountable subset of a second countable space has uncountably many of its limit points

Let $X$ have a countable basis , let $A$ be an uncountable subset of $X$. show that uncountably many points of $A$ are limit points of $A$. This is my attempt: By way of contradiction, assume that ...
31 views

### Proof for “Given any basis of a topological space, you can always find a subset of that basis which itself is a basis, and of minimum possible size.”

The titular statement is used in the explanation of this answer from several years back. I ran across it while puzzling my way through this text, which I bought while I was still in high school and ...
22 views

26 views

### Closure of a set in the weak topology

Let $X$ be a Banach space, $S$ a subset of $X$. What is the closure of $S$ with respect to the weak topology?
35 views

### Jordan curve in $C^2$

Can we find a Jordan curve $\gamma$ in $\mathbf{C}^2$ of class $C^1$ such that the projection to the first coordinate plane divides the plane into infinite components of connectivity.
24 views

### Applying topological definition of continuity to $f(x) = \frac{1}{x}$

I am trying to show that the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined by $$f(x) = \left\{ \begin{array}{l} \frac{1}{x}, \, x > 0 \\ 0, \, x \leq 0 \end{array} \right.$$ is not ...
22 views

### show set of matrices A such that I-A is invertible is open

If $U = \{A\in Mat(2,2) : I-A \text{ is invertible} \}$, how can I show that $U$ is open? I know that the set, say $V$, of $n\times n$ invertible matrices is open. Can I use this fact with the linear ...
74 views

### Prove That $\mathbb{R}^n - \{0\}$ is connected for n > 1

I don't understand where to start proving this since $$\mathbb{R}^n - \{0\} = (-\infty,0)^n \cup (0, \infty)^n$$ Which is the union of two disjoint nonempty open sets, so it can't be connected. ...
26 views

### Topological space $X$ has no subbasis $S$ with property $Card(S)\leq Card(X)$

Let $X$ be a topological space and infinite which has no subbasis $S$ with property $Card(S)\leq Card(X)$. What special properties does it has? For example it's not metrizable. Because the ...
20 views

17 views

23 views

### A partition of unity of a topological space

I have troubles in a little part of the following proposition. Let $(X,\tau)$ be a topological space and $\Im=\left\{U_{\alpha}\right\}_{\alpha \in I}$ an open cover of $X$. If $\Im$ has a locally ...
38 views

### Question Involving Open/Closed Sets [on hold]

Let $f : X \rightarrow \mathbb{R}$ be a continuous function and let $a \in \mathbb{R}$. Determine whether the following statements are true and false. Prove you answer. i) $\{x \in X :f(x) \leq a\}$ ...
38 views

### Embedding $T_{1}$-topological space in $\Bbb{R}^{J}$ [on hold]

Let $X$ be an infinite set and let $τ$ be a $T_{1}$-topology on $X$.Does there exist $J$ such that $(X,τ)$ can be embedded in $\Bbb{R}^{J}$ with product topology and cofinite topology on $\Bbb{R}$?...
21 views

### How is a sequence not converging usually but $I_{\tau}$ converging in this given paper.
I am reading the paper Pratulananda Das and Ekrem Savas: On I-convergence of nets in locally solid Riesz spaces, Filomat 27:1 (2013), 89–94, DOI: 10.2298/FIL1301089D. I am stuck at example $3.2$ ...
I am self-studying the general topology these day and find that the third axiom of the topological space $(X,\tau)$ defined by open set is: For any finite collection of $U_i \in \tau$, the ...