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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.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

57719 questions
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Subspace topology and discrete space

Does there exist $A \subseteq \mathbb{R}^{2}$ with usual topology and $A$ not enumerable such that $\tau_{A}$ the subspace topology in $A$ it is a discrete space?
Jhon Jairo
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Action of $S^1$ on $S^3$

Let $S^3 = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1\}$ and $S^1 = \{z \in \mathbb{C} : |z| = 1\}$. Let $S^1$ act on $S^3$ by $z \cdot (z_1, z_2) = (zz_1, \overline{z}z_2)$. Let $X = S^3/S^1$. Consider $\tilde{f}, \tilde{g}, \tilde{h} :…
user50945
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Prove that the Sierpiński space is a topology

I need to prove that the Sierpiński space, $\mathcal{\tau} = \{\emptyset, \{1\}, \{0, 1\}\}$, is a topology. I have only just started on toplogy, and so far just know the basic axioms. To prove that the union of any collection of subsets from T is…
helpmepls
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Can a continuous topological function be one-many?

Can there exist a continuous topological mapping $f:X\to Y$ which is one-many? I ask this question because if such a mapping exists, then I can see potential contradictions in some theorems stated in my text. I know that one-many mappings aren't…
user67803
4
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1 answer

Continuous map between topological spaces, with two local homeomorphisms is a local homeomorphism

Let $X,Y,Z$ be topological spaces. Let $f:X\rightarrow Y$ be a contionuous map. I also have two local homeomorphisms $p:X\rightarrow Z$ and $q:Y\rightarrow Z$. Such that these functions form a commutative diagram. I need help to show that $f$ is…
user850424
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Every neighborhood of $x_{0}$ contains infinitely many points from a set $A \subset X$ iff $x_{0} \in X$ is a limit point of A.

Let X be a topological space such that for every $x \in X$, the set $\{x\} \subset X$ is closed. I have to prove that every neighborhood of $x_{0} \in X$ contains infinitely many different points from A $\subset$ X iff $x_{0} \in X$ is a limit point…
Laura van Leuven
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Topology of uniform convergence on elements of $\gamma$

Let $\gamma$ be a cover of space $X$ and consider $C_\gamma (X)$ of all continuous functions on $X$ with values in the discrete space $D=\{0,1\}$ endowed with the topology of uniform convergence on elements of $\gamma$. What does "topology of…
TXC
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Proving that there is a path in set.

Given that $\underline Y$ is a subspace of $\underline ℝ^2$ and defined as follows: $$I = [0,1]$$ $$X = (\{1\} × I) ∪ \left(I × \left(\{0\} \cup\left\{ \frac{1}{n} \,\Bigg\vert\, n \in \Bbb N \right\}\right)\right)$$ $$Y = X \setminus \{ (0.5, 0)…
Aelx
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Isolatedness of Lefschetz map - almost there

$f$ is a Lefschetz map on a compact manifold X. And I need to show the Lefschetz fixed point is isolated. I proved that the graph of f is transversal to the diagonal inside $X \times X$, then I don't know how to proceed from here. Thank you very…
1LiterTears
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$\mathbb{Q}$ is not a Baire space?

The condition of 'X is a Baire space': (cite from Topology by Munkres) Given any countable collection $\{ A_n \}$ of closed sets of X each of which has empty interior in X, their union $\bigcup A_n$ also has empty interior in X. So for rational…
whitegreen
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Interesting Normality Condition (topology)

I just got through an inquiry-based course in introductory topology, and I enjoyed the experience a lot. If you're unfamiliar, this roughly means that we were given a text full of theorems and the class consisted of providing as many proofs as…
Eric Stucky
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Lemma 1 - Banach-Mazur Game - Dan Ma's topology Blog

I have been given a presentation for a course based on this particular post in Dan Ma's topology blog. The post itself is very clear and educational, but there is one proof that I'm not sure why it is complete. Lemma 1 has a rather standard…
Deborah
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Determine whether the given points are interior point of the given set

Let $B = \{(x,y) \in \Bbb{R}^2 \mid -1 \le x \lt 2, 0 \lt y \le 2 \} \cup \{(x,y) \in \Bbb{R}^2 \mid 5 \lt x \le 7, y = 1 \}$. Determine whether a point $(0,1)$ is an interior point of $B$. I got a little bit confuse here, since a point $(0,1)$…
lap lapan
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Showing something is an open two cell (Lee page 134)

In Lee's Introduction to Topological Manifold on page 134, it states that the set $\mathbb{B}^2\setminus \{(x,0): x\in[0,1)\}$ is an open 2-cell. However, I'm not quite sure as to how to define a homeomorphism from $\mathbb{B}^2$ onto the set.…
varpi
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2 answers

Application of Urysohn's lemma

I am working on the following hw problem: If we have that $X$ is a compact Hausdorff space, with $\{U_\alpha\}_{\alpha\in A}$, then we can find a finite number of continous functions $f_1,...,f_k$, with $f_i:X\mapsto [0,1]$ such that…
Daniel Montealegre
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