# Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

57,992 questions
Filter by
Sorted by
Tagged with
15 views

### how to find $\text{int}(D)$

On the set of all real numbers with usual topology, let $D = \bigcap_{n=1}^{\infty} \left(0, 5 + \frac{1}{n} \right)$ Find $\text{int}(D)$
10 views

### Product of quotient map $p: X \to Y$ and identity $Id_Z$ is a quotient map when $Z$ is compact and Hausdorff

Let $X,Y,Z$ be topological spaces where $Z$ is compact and Hausdorff. Let $p: X \to Y$ be a quotient map. I want to show that $$p \times Id_z: X \times Z \to Y \times Z: (x,z) \mapsto (p(x),z)$$ is a ...
1 vote
35 views

### Construct a topology $T$ on $X=(-\infty,0]$, such that $X$ is compact? [closed]

Construct a topology $T$ on $X=(-\infty,0]$, which differ from trivial topology and discrete topology, such that $X$ is compact?
27 views

### For all $a\in A$ and $c\in C$, there exists $a',c'\in (a,c)$ such that $a'\in A$ and $c'\in C$. What can we conclude about $A,C$?

The $\mathbb R^n$ space is partitioned into three sets $A,B,C$. Given: $B$ is convex. $A,C$ are non empty. For all open segment $(a,c)$ where $a\in A$ and $c\in C$, there exists $a',c'\in (a,c)$ such ...
• 3,770
25 views

### Algebraic Topology: Deformation Retraction and Quotient Spaces (using Mobius )

Could someone please help me with this question (and its solution)? Thanks!! Solution Part 1 (which shows that [0,1] x {1/2} deformation retract of [0,1] x [0,1]): I don't get how the homotopy gets ...
24 views

### partition of $\Bbb R^3$ by punctured topological surfaces

Consider a partition of $\Bbb R^3$ by topological surfaces $M=\Bbb R^2-\lbrace \mathrm{1~point} \rbrace$. Is it possible to make these surfaces $M$ complete, hence giving a partition of $\Bbb R^3$ ...
• 888
27 views

### topology effect on PDE solution

I'm interested in understanding how the topology of a domain influences the solutions of a PDE defined on that domain. Specifically, I'm curious if there are methods to explore different topological ...
48 views

### how to build a topology T on X such that X is disconnected? [closed]

how to build a topology T on X=(0,5), which difference from trivial topology and discrete topology such that X is disconnected?
28 views

### continuity of a function defined in a compact metric space

Let $K$ be a compact metric space and $S\subset K$ a compact subspace such that $$S \subset \bigcup_{i=1}^{k} B_{s_i}$$ where each $s_i \in S$ and $B_{s_i}$ are open balls centered at $s_i$ consider ...
• 289
1 vote
36 views

### Point-separating fields of clopen sets on compact spaces without Choice

In Matthew Dirk's Paper on Stone's representation theorem there is a proof of Lemma 3.8. If X is a Stone space and F is a separating field of clopen subsets of X, then F is the dual algebra of X; that ...
95 views

• 145
95 views

### A topology different from discrete and anti-discrete, in which every open set is closed, and every closed set is open. [closed]

To give an example of a topology on a set of four points, different from discrete and anti-discrete, in which every open set is closed, and every closed set is open.I have 0 ideas at all, I thought ...
19 views

### prove that every infinite subset of X is dense in X with respect to the cofinite topology [closed]

I know subset A of X is said to be dense in X, if A_bar = X. If F is a closed set in X containing A, then F = X ,How to i show that for every infinite subset also added it is a cofinite topology ...
46 views

### Is metric incompleteness of $\overline{X}$ always witnessed by some sequence in $X$?

This following conjecture seems intuitively true to me, but I have trouble proving it (and I haven't found it confirmed/disconfirmed in the literature or online): Let $M$ be an incomplete metric space....
• 357
19 views

### Is there a classification of connected sets in $\mathbb{R}^2$? [duplicate]

We know that in $\mathbb{R}$ with standard topology, a subset $A$ of $\mathbb{R}$ is connected if and only if $A$ is an interval (including points, rays, $\mathbb{R}$). Is there a classification of ...
• 1,179
1 vote
26 views

### Singletons in regular spaces

I need help with a statement. Let $X_1$ and $X_2$ be topological spaces and consider $X = X_1 \times X_2$. Suppose the product space $X$ is regular, that is, for any closed subset $C \subset X$ and ...
• 101
50 views

### Give examples of open and closed sets

Task: Let the topology $T = \{\emptyset; \mathbb{R}; \left(-\infty, 1/2^{n}\right], n \in \mathbb{Z}\}$ be given on the line $X=\mathbb{R}.$ Give examples of: a) an open but not closed set b) a closed ...
1 vote
27 views

### A certain distance-related map is well-defined

I am trying to understand this answer. Context: we want to show that there exists a retract $C\to A$ where $C$ is the Cantor set and $A\subseteq C$ is a nonempty closed subset. They suggest to take a ...
40 views

### Find a bounded real sequence, whose range has exactly one accumulation point, but which is not convergent [duplicate]

Example of bounded sequence on $\mathbb{R}$ s.t. the set $\{X_n : n \in \mathbb{N}\}$ has exactly one accumulation point but $X_n$ is not convergent My thoughts: Since Xn is bounded it has ...
42 views

### Prove that open balls are preserved under scalar multiplication

Suppose we have a random ball $B_r(x)$ in $\mathbb{R²}$, and we use the vector space structure and scale this ball down by a non zero scalar $a$. I believe that, $\frac{B_r(x)}{a}= B_{r'}(x')$. My ...
• 11.7k
37 views

1 vote
51 views

### A full embedding functor from $Sob$ to $Frm$?

In the following passage, the authors Picado and Pultr say the functor is a full embedding then they say that it would be a full embedding under some condition. Any explanation?