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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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The Real Projective Plane is a 2-dimensional manifold

To demonstrate this do I just have to show that $P^2$ is Hausdorff and locally Euclidean? I can show that the space is Hausdorff but I'm having a little trouble demonstrating that it is locally Euclidean.
user62931
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When does $\overline A = \bigcap\{\overline U:\ U\text{ is a nbh of }A\}$ hold?

Let $X$ be a topological space. For any $A\subset X$ we have $$\overline A = \bigcap\{ C:\ C\text{ is closed and }A\subset C\}.$$ Suppose that $X$ is Hausdorff, is it true that $$\overline A = \bigcap\{\overline U:\ U\text{ is a nbh of }A\}?$$ …
BigbearZzz
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Why a circle with a hole in it is not compact?

Take the real plane $\mathbb R^2$ with the standard topology, and take $A\subset \mathbb R^2$ which is a closed circle minus a smaller closed circle: I am trying to understand why this set is not compact, the hint is to use the Heine-Borel theorem.…
Jacob Morales Gonzalez
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How can a continuous function map closed sets to open sets (and vice versa)?

Definition of continuity: A function $f: X \to Y$ (where $X$ and $Y$ are topological spaces) is continuous if and only if for any open subset $V$ of $Y$, the preimage $f^{-1}(V)$ is open in $X$. Now, if $U$ is a closed subset of $X$ (meaning that…
Zelyucha
  • 191
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Genus in topology

Can somebody provide a formal definition of genus in topology? I find it difficult to imagine what genus is. For example, objects of genus zero are the ones that homeomorphic to a sphere? What about higher genus?
mrk
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Maximal set with respect to the finite intersection property

Let $X$ be a Hausdorff space. Let $\mathcal{D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. That means that $\mathcal{D}$ is a family of subsets of $X$ that has the finite intersection property…
user44069
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Why is $(0, \infty)$ an open set and $ [0, \infty)$ a closed set?

Specifically, I don't understand how to think about the infinity part. Closed and open seem to have pretty intuitive definitions when not considering infinity.
user3180
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Subspace topology is the unique topology that satisfies the Characteristic Property

Lee has as an exercise that the subspace topology is the unique topology that satisfies the characteristic property which is for $S\subset X$ a function $F:Y \to S$ is continuous iff the composition $i_S\circ F$ is continuous. I think that he means…
Anthony Salib
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Is the two-point compactification the second-smallest compactification?

We know that the Alexandroff one-point compactification of $\mathbb{R}$ is in a precise sense its smallest Hausdorff compactification. Is the two-point compactification of $\mathbb{R}$, in a precise sense, the second-smallest? In other words, given…
SSF
  • 905
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Show composition mapping is continuous with compact-open topology

Let $X$ be a compact Hausdorff space, and $H(X)$ be the set of homeomorphisms from $X$ to $X$, with the compact-open topology. Prove that the mapping $h:H(X)\times H(X)\rightarrow H(X)$, $h(f,g)=f\circ g$ is continuous. Note, if $C(X,X)$ is the set…
The math god
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Subset of $\mathbb R^n$ homeomorphic to sphere?

Let $C$ be a subset of $\mathbb R^n$ with the following properties attached to it: Convex Compact Non-empty interior Is the boundary of $C$ homeomorphic to the ball of dimension $n-1$? Why? Thanks in advance!
Ulibniss
  • 127
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Topology From Sequences

Let $X$ be a set and $f:X\to\mathcal{P}(X^\omega)$. Under which circumstances is there some topology $\tau$ on $X$ such that $f$ maps each point $x$ to the set of converging sequences in $\tau$ with limit $x$? The finest topology which realizes…
fweth
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A subbasis $S$ for a topology on $X$ is a collection of subsets of $X$ whose union equals $X$.

Munkres definition says the following: A subbasis $S$ for a topology on $X$ is a collection of subsets of $X$ whose union equals $X$. The topology generated by the subbasis $S$ is defined to be the collection $\mathcal T$ of all unions of finite…
STEMExchanger
  • 85
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Mistake in Dugundji chapter IX section 11 example 3?

A family of pseudometrics defined on a set gives rise to a uniform structure on that set. Moreover (up to uniform equivalence, anyway) every uniform structure arises this way. Let $A$ and $B$ be families of pseudometrics defined on sets $X$ and $Y$…
Mike F
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Relation between the notions connected and disconnect, confused

In the textbook "Topology without tears" I found the definition. $(X, \tau)$ is diconnected iff there exists open sets $A,B$ with $X = A \cup B$ and $A \cap B = \emptyset$. In Walter Rudin: Principles of Analysis, I found. $E \subseteq X$ is…
StefanH
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