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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.

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How can I prove $\tau$ is a topology on $\mathbb{N}$

Consider the set $X = \mathbb{N}$, and let $\tau$ be the collection of all subsets $A \subset \mathbb{N}$ for which $\mathbb{N}\setminus A$ is finite, along with the empty set. I want to show $\tau$ is a topology on $\mathbb{N}$. I know that the…
user15412
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Determining if following sets are closed, open, or compact

I'm doing some topology now and I'm honestly very confused on how to formally say that a set is open or closed or compact. Like especially with open sets, the proof for it (balls must exist of radius $r$) seems very arbitrary (since the radius can…
Anthony
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Why should $2^{\aleph_0}$ be the maximum cardinality for regular spaces with countable bases?

A regular space $X$ is a topological space for which one point sets are closed and for every $x\in X$ and for every closed set $C\subset X\backslash \{x\}$, there exist disjoint open sets containing $x$ and $C$, respectively. Assume $X$ is regular…
confused_wallet
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Is it possible to partition a topological space into nonempty disjoint closed subsets?

Let $X$ be a topological space containing at least 2 elements. Is it true, in general, that there exist 2 disjoint closed nonempty subsets $A$ and $B$ of $X$, such that $$X = A \cup B?$$ Is it false, in general?
Andrey Portnoy
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Showing that a locally finite collection preserves the closure

I managed to show (a), but am stuck at (b). I think I have to manipulate the definition of local finiteness but cannot find a way. Could anyone please help me with (b)?
Keith
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Visually, why is the 2-sphere $S^2$ not contractible?

In topology, can someone please describe why the sphere $S^2$ is not contactable? Surely it can just 'shrink' to a point?
user53076
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Is $\{(0,0)\}\cup\{(x,\sin{1\over x}):x\in\mathbb{R},x>0\}$ path connected?

Is $\{(0,0)\cup\{(x,\sin{1\over x}):x\in\mathbb{R},x>0\}$ path connected? I think it is path connected if we neglect the point$(0,0)$ it is as we can define a continuous function easily from $[0,1]$ but if we included the point $(0,0)$, then any…
Mathematics
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Why is $(a,b)$ open on the lower limit topology of $\mathbb{R}$?

In the book of General Topology by Munkres, on page 104, it is given that However, as far as I know, the lower limit topology $\tau_l$ corresponds to the intervals of the form $[a,b)$ where $a < b$. So how can $(a,b)$ be open in this topology?
Our
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Can a path avoid a dense set?

My actual question is related to an unusual circumstance in a game which I am playing. We are trying to move one point-like object from one part of the moon to another without passing through another player's land. However, due to a loophole,…
Eric Stucky
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Some T2 spaces must have a small dense?

If a Hausdorff space $\ X\ $ admits a dense subset $ A \hookrightarrow X\ $ such that $$|X|^{|A|}\ =\ |X|$$ then indeed $$|X| \leq |\text{End}_{\text{Top}}(X)| \leq |X|^{|A|}\ = \ |X|.$$ It is the case of $\mathbb{Q} \hookrightarrow \mathbb{R}$.…
Ivan Di Liberti
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$f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous

Possible Duplicate: No continuous function that switches $\mathbb{Q}$ and the irrationals Let $f: \mathbb{R} \to \mathbb{R}$ be function satisfying the two conditions: $f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q})…
Sayantan
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If $ \cup \bar{A_{\alpha}}$ is closed on $ X $, then $ \cup \overline {A_{\alpha}} = \overline { \cup A_{\alpha} }$

Let $ (X, \tau) $ be a topological space and $ \{A_{\alpha}: \alpha \in I \} \subset P(X)$. Verify or disprove the following: If $ \cup \overline{A_{\alpha}}$ is closed on $ X $, then $ \cup \overline {A_{\alpha}} = \overline { \cup A_{\alpha}…
u0tf1s
  • 397
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Can all surfaces be turned inside out?

I saw this YouTube video claiming that spheres, double torues, triple toruses, etc. can all be turned inside out, but what about other surfaces? Are there any surfaces that can't be turned inside out?
user80458
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Why do I think Lebesgue’s number lemma is wrong...

Counterexample: $[0,1]\times[0,1]$ with induced subspace topology from $\mathbb{R}^2$ is compact. The open cover $\mathscr{U}$ is just the two circular sectors. When we look at the up-left corner and down-right corner, it fails - there is no such…
Upc
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Maximal compact topology iff compact sets are closed

The theorem says: A compact space $(X,\tau)$ is maximal compact (i.e. no strictly larger topology on $X$ is compact) if and only if every compact subset of $X$ is closed in $\tau$. For "if", I prove by contraposition. Let $U\subset X$ be a compact…
Sid Caroline
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